[ap27]

There are two things to look at when going for a battery. Voltage and capacity. Voltage will determine your rate of fire, and capacity will determine how long the battery lasts (and how big it is).
For a stock TM 552 you should look at an 8.4V or 9.6V NiMH battery. Capacity can be whatever is suited to your needs. However keep in mind that the cells get larger as you go higher in capacity. Usually a 2000mAh battery should easily last you through a long game. It really depends on how trigger-happy you are.
If you go with a Lipo, keep in mind that they require special attention in terms of charging/balancing. A 7.4V Lipo would give you a rate of fire similar to an 8.4V or 9.6V is most cases. 11.1V Lipos would probably give you a ridiculous rate of fire...but it will wreck your gearbox without significant upgrading.

As for the upgrades, FPS isn't everything in terms of performance. Most outdoor fields limit FPS to 400, and indoor ones limit it to 350. This is a limit though, and not a goal. Stock TMs usually shoot around 280-300fps, which would be plenty for indoor games and reasonable enough for outdoor games (just use slightly heavier bb's for better performance, 0.25g or 0.28g would be nice).
In terms of upgrades, you could do a spring swap to bring up the fps. However using significantly heavier springs would probably warrant upgrading other components in the gearbox. If this is your only gun you should aim for around 330-340fps, which can be achieved with an m100 spring. If you don't have prior experience in working with the gearbox, I recommend finding a gun-doc or someone with experience to do the swap for you.
A lot of people will also recommend you get Age-Verified on the forums here. Once we know you are above 18, we can better direct you to find guns, parts and the people who can get them or fix them for you.

If you plan on just using the gun for 'playing with friends', please keep in mind that airsoft rifles are firearms and will be treated as such by the police or other bystanders. Do not play in public areas, in your backyard or in an empty field. Most cities also have by-laws that make it illegal to discharge a firearm within city limits.