unfortunately, the games only have 1 save file, so you can't start a new game without deleting the old game first. Once you beat the main story, there is a bunch of post game stuff you can do. You are free to explore all the places you have been to at any time, although there is little to do in most places post game.
Other than that, you can train a competitive team and get involved with online battling. SS and B2 also have battle facilities which have various competitions where you can win points to purchase TMs (moves) as well as special items.
You can also rematch the final boss in any Pokémon game (well, okay, you'll be rematching a different character in Black 1 and White 1, but same basic result.) The game will pretend you never completed it as soon as you start challenging the "Elite 4" (basically, the penultimate bosses). Sometimes, the Elite 4 and final boss will have upgraded teams when you rematch them, but not always.
The world outside the Elite 4 and final boss changes a little in the postgame, though, and new areas open up (how many depends on the game - the original games only had one postgame-only area, while the second-gen games and their remakes have an entire (slightly-dumbed-down) second region to explore - which was actually the region of the first-gen games.) --- Creator of Jay's Journey (see quote!) "It's not ten years old! Therefore, it sucks!" - Nostalgia whores everywhere
Soul Silver's postgame: - 8 New Badges - A new region (Kanto) - Battle against the hardest trainer (Red) - An area where you can rebattle all 16 gym leaders. - Battle Frontier, which is 5 different facilities that act as the ultimate challenges that differ in gameplay. It has a place where you use only one Pokemon and choose your opponent's types, a place where you use and trade rental Pokemon, a place where your team is boosted or weakened by a roulette, a place where you have to finance extra levels and held items for your team, and a regular battle facility where you have to win streaks of battles. It's a lot of fun and quite challenging. - Of course, more Pokemon to be found.
Black 2's postgame: - A new section of the map opens up, giving you another 3-4 towns. - Battle Subway (available pre-postgame but only really feasible afterwards) where you have to win streaks of battles. - Pokemon World Tournament, an arena where you face off against all 40+ gym leaders in the series, some champions, and some NPCs in a bracket style tournament. Good fun to be had here. - More Pokemon to catch and train, with several legendaries becoming available. - Online battles where you go head-to-head with other Pokemon players all over the world.
Overall, Soul Silver's postgame is boosted by the Battle Frontier greatly, but Black 2's adds a competitive online touch. --- The Official Bulbasaur of Pokemon X's board Bowser Jr, T Zelda, Ridley, Palutena, Pittoo, Isaac, Shulk, Mewtwo, Robin, Balloon Fight, Crono, Mii, Megaman, Neku