Whatever sounds good to you would be my recommendation. What sounds good to me may not sound good to you. Also, you kind of have to know what system you have now, whether or not you have an amplifier now or are going to add one and whether or not you want a 2-way or 3-way speaker, component system, etc...
You need 3.5" for the dash, (which you may or may not have now...you didn't indicate what head unit you have), 5-1/4 to 6-1/2 for the front doors (with adapters) and 6x9's in the rear.
You may or may not need an amplifier depending on what system you have. You will also need a LOC if you are planning on adding speakers where you don't have speakers now and to do that you'll also need an amplifier.
So there's a lot to kind of think through but the first thing I'd do is go listen to some.
1. does the current setup sound okay, but maybe not enough punch or loudness?
2. does it have enough punch or the sound isn't what you'd like it to be? not enough bass or treble, etc?
If answer 1 is yes, then amplifier upgrade is more what you are looking for. If answer 2, then definitely you are on the right track for speakers.
To help narrow things down here are the stock sizes:
As has been said already, you don't have to go with the stock size in the door. You can use the 5 1/4 or 6 1/2 provided you use an adapter to fill out the 6x9 opening.
Maybe consider a set of components for up front. You mention $300. That should be enough to get a good set such as polk audio's mobile monitor series, for example.
And it may matter whether you have the stock amp or not as the amp is optimized for speakers with 2 Ohm impedence. It's not as critical as some consider it to be, but if you went with aftermarkets that have 4 Ohm impedence, you would lose some output volume.
Aftermarket speakers usually have better sensitivity so you don't lose as much, but it is
something to consider.
As has been said, listen to some speakers in the sizes mentioned above, then listen also to a good component set and see what sounds good to you.