The "standard" main game is included in the link from Monster Iestyn. It features plenty of levels. I personally advice first playing as Tails, then Knuckles, and then Sonic & Tails, due to (in my opinion) Tails being easy, Knuckles sort of being medium, and Sonic being hard. (But remember there are several levels missing in the middle of the game, so after a few zones, the difficulty curve will quickly rise.)
Make sure to set up the controls to your liking as soon as you start playing, regardless of who you play as first.
Note the game is designed for being played with a keyboard and mouse, so keep that in mind, and don't try to make a keyboard-only control scheme first time around (unless you don't have a mouse beyond the touch pad of a laptop).
I personally play with a gamepad, however it's mostly best played using a keyboard and mouse, and it's tough to set up a gamepad first time around.
Once you beat the "standard" main game, try going back to this forum/message board, and go to the Releases sub-forum.
Mods are files which change the game wildly, some times being a big map pack, other times being read/write functionality no add-ons really utilize yet. You know, just things that change the game a lot.
Levels are usually 1-3 levels or such, I think.
Characters are... well, characters you can play levels with.
Lua is for scripts that add or change stuff about the game, some times only being useful for mappers, other times changing core gameplay a lot.
Miscellaneous is pretty much anything that doesn't fit in the other categories.
Regardless of which section you explore first, you have to put the Wad or Lua file or such in SRB2's installation folder (or a sub-folder there-of), and then type "addfile [filename].wad" (or .lua) in the console to add it into the game.