On the main menu, you will have a Select Character option. That lets you switch the active single player save. Once chosen, select Continue.
If you join another host's game, you can also choose which of these saves (characters) to play as in the first screen it shows you after you connect to the server, via the Select Character menu option.
Any progress in single or multiple is saved automatically per save. If you played co-op on the first save and progressed through the missions with others in co-op on their save (mission progression), any you complete with them will be saved on your character and you can't "go back" without creating a new character.
See How does mission progress work in multiplayer? for a description of how MP progress works when playing on other's hosts.
In general, your build in early game does not matter much. Just make sure you phaselock those mutants as often as you can. If you are still uncomfortable with your build, try shifting into Harmony to heal better.
As for money, no, you can't store it in the bank, but just stop caring! Every time you die, you lose 7% of your current money—the less you have, the less you should be afraid of dying. I've never had problems with money—even if you miss that rare gun in the shop, it is not like you won't find a better one soon.
There even is a reason and a way to keep you money low voluntarily: Slot machines. There's a low but reasonable chance to win Eridium there, and as cost of playing rises with the level of main quest, there's zero point in saving for later—money inflates, Eridium doesn't! My personal way of playing is to burn all my money in slots after each trip to battlefield. Longest forays usually net me mostly two to three "rounds"—that is, gambling sessions that end in full inventory (of course, buy inventory upgrades first).
Gamble!
- No money - no fear of death!
- Accidental good gun!
- Average of 0.1916 bars of Eridium per pull!
- Eventual "Jackpot!" badass challenge!
The only thing you need a lot of money for is "For the Hoard!" achievement - and accumulating 3,000,000 is fairly easy even at the end of Normal, and trivial at the end of TVHM.
Best Answer
The best bosses to farm for items are BNK3R, Warrior and Terramorphous. You can't fast travel to these locations, so what you need to do is kill the boss, quit out of the game and then restart the game. The boss will be back and you'll be right at their front door instead of having to run the respective gauntlet before each boss.
Note that Orange/legendary drops are extremely rare except from certain bosses (even then orange drops are absurdly rare. Killed BNK3R 20 times this weekend and got squat). Farming is an effective way to get purple/blue weapons from any boss, but you'll need to target your farming if you want the legendaries.
Also note that Terra always spawns at level 52, so if you aren't level 50, don't bother. See also my guide on killing the crap out of Terramorphous. Similar to the advice in that question, The Bee is an effective way to farm any boss; it used to be insanely easy, now it's just significantly helpful. The Bee has been nerfed (more specifically Amp Shields in general) so the boost is less significant for multi-projectile weapons, so if you're using the Bee, try a single bullet per shot, high rate of fire weapon like a Spinigun instead of a shotgun.
BNK3R in particular is easy to farm for Assassins since the Bore skill is bugged for him and sometimes he'll take repeated hits from it (with increasing damage) and die in very few hits (or one).
However, regarding farming before playthrough 2...Don't farm before playthrough 2. Actually you need to be in playthrough 2.5 (e.g. complete playthrough 2) to get the best items possible. Don't farm until your level is maxed out; unlike Borderlands 1, damage follows an exponential scale in Borderlands 2. This means almost immediately after playthrough 1, orange level 30 guns are total crap compared to green level 35 guns. Ditto even up to level 45 guns. Everything is crap until you hit the level cap, just play normally until then and don't worry about farming.