You're getting a lot of random advice. Unfortunately, threads like these invite the, "I shoot with a ______ and therefore recommend it."
And beware of suggestions like, "I shoot with a powershot and it's great" -- ALL Canon P&S cameras are called Powershot, some are pretty good, some are pretty awful.
All cameras are about compromises. Regardless of budget, even if you are spending $10,000 cameras, you have to make compromises. (the 2 biggest compromises are price and size... To get "everything" you need to spend a lot, and it won't fit in your pocket).
So you need to break down your "needs" (not that a camera is ever an actual need) versus your wants.
Price... You said $300... Do you "need" to keep it under $300, or is that your preference, with an ability to go higher if necessary?
Long distance -- How long do you *need* to go, versus *want* to go? This is important, because long zoom point & shoot cameras often contain a lot of other compromises, so you are giving up a lot, to get that "long distance" shooting. So do you need to be able to shoot across a room? Do you need to shoot kids 10 feet away on a playground? Do you need to shoot all the way across a baseball field? Do you need to shoot birds in flight? How far do you *need* to shoot, and then how far would you *like* to shoot? (Perhaps you would like the ability to shoot kids playing soccer from 100 feet away, but you rarely take such pics and don't really need it).
Indoors -- As I mentioned above about compromises, for the most part, most P&S cameras that can shoot a long distance, are truly stinky indoors. So do you need to shoot indoor, or is that a want? Do you need to shoot indoors without a flash, or are you okay always using a flash? Really, 95% of P&S cameras stink indoors without a flash, but there are a handful of good ones (that cost a bit more and/or are a bit bigger and/or sacrifice the long distance).
Portability -- How small do you need it to go? (Again, versus want). Need it to fit in a pants pocket? A large jacket pocket? Need it to fit in a small handbag? Need it to fit in a medium sized purse or backpack?
With answers to these questions, intelligent recommendations can be made. Based on the limited information I have so far, I would recommend
http://www.amazon.com/gp/product/B0..._m=ATVPDKIKX0DER&pf_rd_r=0NKR6VQW3F86S2ZMTT2F
It has a fairly long zoom range. It's not too huge though it isn't tiny. It has a "fast" lens, so it's not totally useless indoors. If you are willing to buy grey market, it is right around $300.