If I am one card away from a straight or flush after the flop, should I keep chasing it even if others are betting? Then fold after the last turn if I end up having nothing? I am pretty new to poker and right now my biggest struggle is knowing what to call/hope for a big hand and what to fold. Also, does anyone have a good strategy for calling or raising when you have two pair? I have gotten excited with a 2-pair a few times and staying in and losing to three of a kind.
As far as calling with a straight draw or flush draw, I suggest familiarizing yourself with the concept of Pot Odds. It's an important factor behind making the decision to call or fold with a draw, but it's also something that drives the decision-making behind almost every hand. If you search this site for Pot Odds, there are quite a few questions with answers about it; alternatively, there are many poker books and instructional sites that explain it.
As you're new to poker, one of the things you'll learn about it (which is enticing to some, and frustrating to others) is that there's rarely an easy, "correct" play--there's many factors to take into consideration. For example, even two different hands where you have a flush draw can be completely different. If you have the ace of a suit with the flush draw, that's a stronger draw than when your highest card is a 6, because the 6-high flush can still be beaten if you make the flush, whereas the ace-high flush could be unbeatable. A flush draw when there's no pair on the board is better than a flush where there is a pair on the board, because if you hit a flush when the board has a pair you can still potentially lose to a full house.
As for what to do with 2-pair, again every hand is different and there's lot of things to consider which you'll learn with time if you keep studying. Also, like the flush draw example, not every two pair is equivalent in strength. Having 75 when the board is Ace-7-5 is not as strong as having Ace-7 on the same board--when you have 75, anybody with just an ace can end up getting a better two pair, whereas having Ace-7 is less likely to end up getting beat. Also, you have to ask if there is a flush or straight possible--if so, your hand isn't quite as strong.
Very generally, two pair is usually a good hand to have after the flop and you should be comfortable betting with it. But when people start raising your bet and/or if a lot of better hands are possible (higher two pair combinations, straights, flushes), you need to slow down and be cautious of putting in too much.
Way different questions.
2 pair is a good made hand. With 4 outs to be a very good hand (full house). If one pair is on board you cannot count that as your opponent also has that pair also.
Chase a 4 (straight) or 9 (flush) outer draw is way different. Too much for one question. Lots of math available free on the Internet or books (check out Sklanksy).
On the flop 4 outs is 41 / 4. You should call if you are getting 10:1 - call a bet of 1 into a pot of 10. Any decent bet will price you out.