There are two types of 1 vs 1 games I'll cover here, earthquake and normal. Dogfight is more or less who has more scouts so I won't talk about that. In both types you should start out as communist.
EARTHQUAKE
For earthquake, snipers are the best unit because of their range. They can fire without being hit at groditz, hover tanks, base cannon, and most importantly, other snipers (if you micromanage them well).
Start off with a bank. Then look at the opponents base. If they are building an outpost (maybe a rush) then build an outpost. If they are building something else, then make a hospital.
Next make an outpost if you have not done so already. If there is still no threat, make an armory. If there is, upgrade to barracks.
As soon as you begin building the last building, switch armory to plan reinforcements.
When you have everything completed, make medics and snipers at the same time. The rate of production ratio should be your total number ratio as well.
March them towards the enemy and take the middle ground. If you built correctly, you will have more snipers then your enemy and your medics will help even more. If they have groditz or hover tanks, just walk in, snipe once, then retreat. By the time you have retreated back to your base, you should have devastated the enemy army. I have done this against 10 advancing hover tanks without losing a single sniper.
With your enemy crippled, finish them off. Make sure you have lots of snipers though. Once they attack the base, enemy units can now hit them.
NORMAL
The most powerful strategy here is Hover Tank and Sakata combo. About 80% of people tech forge, 15% tech outpost, and the last %5 use this hover/sakata combo.
The important thing to remember is that your hover tanks are gonna be your firepower, and sakatas are secondary, only for anti-air defense. Hover tanks destroy infantry, chronite, sniper, groditz, spiders in one hit, and gladiators in 4 hits (but the hover has far greater range).
Bank, Armory (unless rushed), forge, then upgrade. Get sakata defense right away because your opponent probably did forge and is rushing you with scouts or phantoms. Rarely they will attempt a roman only rush this late in the game because a single phantom can take care of them all.
Next build outpost and then tech barracks. By this time, you should be heavily attacked so save the energy needed for the manufactory into building sakatas. After you're safe, upgrade manufactory, switch armory to energy and start spamming hover tanks.
Keep building sakatas and if you find that you're low on energy, demolish your bank in favor of an energy/solar grid. None of the units you will build now needs money.
Then make two lines of hover tanks with sakatas in between. Advance at the enemy base and type "gg" into the public chat.
Play me sometime. As of August 30, 2009 I have 300 games played with 80% win ratio. In a few weeks this will be a worthless figure but its a decent stat right now.
