Step 1: Purchase a plantation in the Spanish province (A traditional harvesting grounds for recruiting of soldiers. Few citizens, mostly considered ‘backwater’ at the time.)
Step 2: Introduce the horse-collar, the plow, ‘terra preta’ (tilling of charcoal), fertilizer (easiest in this case: baked pig manure), insecticide, pennicillin, and wind-mill-powered pumped irrigation. These things together will result in massive harvests at low costs. They will also ensure that my peasants’ children don’t die, and are well-fed. My wealth will after one or two years of this expand massively. The absence of disease will also enable me to convince the local population that I am favored of the gods.
Step 3a: Utilize newfound wealth and the introduction of the pulley, the A-Frame, and the leaf-spring to build a fleet of Conastoga wagons for transport of crops and local goods at faster rates than anyone else could accomplish. Additionally introduce the water screw and associate it with the aeolipile (already-existing but unutilized historically) for water transport. Introduce distilling for alcohol (another trade good) and also for methalization of olive oil for biodiesel for fuel for simple steam-turbine engines (aeolipile --> low-efficiency turbine pretty easily). These will then be the power-train for both Conastogas and for boats, again increasing mobility of my products.
Step 3b. Reinvent the Girandoni air rifle. Work with metalsmiths under my control to do so. These rifles are 20-shot .46-caliber air-powered rifles first used for military applications in the 1780′s. While technically difficult to create, the metalsmiths of Rome could accomplish this. These smiths would be kept under death-watch; 24⁄7 watch of guards dedicated to keeping the secret by killing any smith who was about to be captured or defected. Said smiths will otherwise be granted lives of utter luxury. Equip motorized Conastogas w/ flame throwers. Recruit/conscript local peasantry into service in newly minted mobilized infantry.
Step 3c. Aeolipile/turbine biodiesel engines provide sufficient weight/power ratio for ultralight aircraft. Constructing a small core of these and training young-teen pilots will enable scouting profiles unlike that of any people to have come before. These will not, however, be available until the third or fourth year, as building the frames will require bamboo grown from imported plants. (Some variety of phyllostachys, likely.) Amongst their other purposes, the bamboo shoots will be used to construct lightweight frames for ultralight craft. Coupled with simple radios, these would prove invaluable for distribution and acquisition of information. Between the mobilized infantry and ultralight air superiority, even a small force—say, four groups of ten wagons w/ twenty soldiers each, and five ultralights—would be an equal match to a single Roman legion despite being outnumbered several times over. (less than a thousand men vs. 11,000-16,000 men.) (Logistics are also an issue. But it’s relatively easy to beat that problem just by using blown-glass bottles and corking them while boiling.)
This lets my troops move faster and strike harder than any unit of traditional troops—several times over.
Step 4. Expanding my scope of influence. Establishing myself as a military force to be reckoned with, I will position myself for further trade for economic expansion before initiating aggressive hostilities. ( I want to wait until the Emperor’s people make the first move to ‘suppress’ me so when I come as a conqueror I have a plausible excuse; I was simply ‘defending’ myself against a mad emperor.) In so doing, I will introduce arabic numerals and double-entry book-keeping. This will enable me to engage in fractional-reserve banking and begin moneylending to foreign persons, thereby expanding my own wealth and influence. I will further develop a telegraph system to permit communication amongst disparate parties (and distract from the existence/use of radios, whose crafting will also be another ‘secret of the gods’, similar to the making of the rifles. Unlike the rifles, the radios will be far easier to be stolen. Need to plan ahead for encrypted communications.) Abaci, too, will be introduced for accountants.
At this point it is four to five years into “the plan”. I now own a massive swath of plantations, each of which are producing yields at unheard of rates. I will have introduced greenhouse farming for things like strawberries or rice for sustained harvests as well. My fleets of motorized wagons and steam-boats will be making an expanding circuit of influence. The peasants under my care will be practically immune to disease (vaccination and antibiotics.) -- an intentional campaign of propaganda (this will require introduction of the printing press and advances in grammar such as punctuation and word spacing [ !! ], as well as use of radios w/ loudspeakers for ‘town-square addresses’. ) will ensure that the people are suitably convinced that this is because I am the favored son of Vulcan, god of the forge. My troops will be invincible, and terrifying. (Ultralights could also be used to deliver non-inconsequential quantities of ‘greek fire’ (gelatinized biodiesel in this case) to ensure that enemy fortifications are essentially indefensible. This will spread fear and further reinforce the Vulcan-myth; I harness fire to move vehicles through the very air itself, and rain fire down from above.)
Years six through ten depend on unforseen variables for whether there will be protracted suppression of luddites and conservatives, or if my reputation precedes me sufficiently that Rome begs me to induct her into my Constitutional Republic of Aligned City-States.
Years 6+ will also involve the introduction of schools for training the children of regions in my control in basic rationality arts; skepticism, falsificationism, logic, mathematics, etc., etc.. They will also introduce further socioeconomic and technical advances that require a more advanced base than 0-year Rome possessed. These include factories, vacuum tubes (and through them early-tech varieties of computers), birth control and other forms of concerns for quality-of-life, and various forms of chemistry and so on.
Here’s why nowhere above did I mention gunpowder: it’s too easily stolen a technological advantage, and explosives in combat are too effective at reducing the advantage of disparity in technological gaps. One guy on horseback w/ a tube-launched gunpowder rocket could maul a Conastoga wagon; one hundred men with such weapons would eliminate the combat-efficacy of my above mobilized infantry. Mortars would be even worse. The same goes for the ultralights; they are defenseless against massed gunfire, AA weapons like rockets, and the like.
However—knowing this myself, I can keep that in reserve for the second-generation of technological-advantage combat revolution. Motorized bicycles or bronze-plated half-tread conastogas require more technical prowess initially but make for just as effective a force, especially if they too are using light cannon, rockets/mortars, and the like. Ultralights equipped with unguided mass-fire rockets are less valuable but after 6-10 years would be more expendable. And their value can be expanded through use of dirigibles as carriers. (Extended range plus heavier craft w/ more-powerful motors.)
Through a careful management of new agricultural, medical, economic, industrial, and military technologies as above, conquering Rome would be easily achievable within 5-10 years at the absolute most.
If one wishes to gamble more, one year would be the minimum necessary to assemble an elite corps of four hundred Girardoni air-rifle troops (coil-spring-driven boltguns would make clumsy but effective pistols for close-range combat in a pinch) riding motorized Conastogas equipped w/ flamethrowers, assuming we acted piratically, canniballizing a few small towns in the countryside to sustain ourselves through the build-up period. These would then be used to launch a Hannibal-esque strike upon Rome itself, designed to assassinate the Emperor and place myself upon the throne.
EDIT: I just realized that it’s possible to hand-solder transistors from raw materials (As well as capacitors and other similar parts). This means that it is possible to create electronic components using Roman-era tools to build the tools necessary to assemble said electronics. While they wouldn’t be up to snuff for modern designs, they would certainly be sufficient for crude robotics. Including ‘rapid’ prototypers. I was already failing to fully utilize my time during the aggri-business expansion (year 1) -- this would adjust for that. I could build machines to handle much of the industrial bootstrap up. (Gun manufacture in particular.) I wouldn’t even need blacksmiths for the guns or turbine-engines. Sintering, casting, and assembler-bots would be capable of producing the equipment as needed. (As was also noted to me by another person; clockwork automata can achieve complex if stereotyped patterns of behavior up to and including fine manual dexterity actions like uncorking and pouring bottles of wine. The industrial applications of this are immense. I wouldn’t even need gunsmiths or engine-smiths; I could instead automate the nearly entire process.)
Neither of which are known to the Wizarding world, as evidenced by the Occlumency teacher’s reaction to his discovery of it. (and his discovery of it, and his discovery of it… :) )
Your assessment of the Wizarding World’s evaluation of the Muggle world. To the supermajority of Wizards, science is a total unknown. Economic and sociopolitical theory are terms they’ve simply never heard of.
They are isolated and effectively are like the apocryphal Chinese Emperor who burned his fleets because there was nothing left to discover; or the equally apocryphal Patent Office official who wanted to close the Patent Office in the 1800′s because there was nothing left to invent.
So basically what you’re seeing is what’s called “hindsight bias”. It is obvious to you, who knows what “Muggles” have, that the Wizards are vastly disadvantaged here—insanely so—but remember that as further demonstrated by Draco’s total ignorance of Man’s visit to the Moon, Wizards believe Muggles are “wallowing in the mud”. The idea that they might LEARN from Muggles is actively suppressed by a concerted political campain by a powerful and long-standing major political faction.