JamesRifle, that is a tough one. First let me say that when I started here I did not want to spend much on air rifles. I wasn't really sure I wanted to throw a huge chunk of money into this hobby. So, to that end, I started with some very inexpensive springer rifles made by Benjamin. I did some work to enhance the trigger, and that helped. I ended up replacing the Benjamin factory provided scopes, as factory scopes are usually of a poor quality. The factory scopes also had problems with the zero drifting on them, so each day meant tweaking the zero on the scope. By the way, this is typically true of any scope which comes bundled with an airgun of any manufacture. After the critical faults were dealt with, the guns shot fairly well. Please keep in mind I started shooting air rifles in the 1960s, so I understood how to deal with the reverse recoil of springer rifles. A newbie must master the special hold most inexpensive air rifles require to be anything approaching accurate. That said, I would not take any of my springers to a match competition. The good news is there are a lot of new PCP air rifles coming out which also fit your budget (okay, you have to spend a bit more to get an air pump). Let me give one example: The Benjamin Maximus with an air pump bundled costs $350, but once you apply the "AGNATION" discount code at check out, it drops to $262.
The link for the air rifle is here:
http://www.crosman.com/airguns/air-rifles/maximus-kit-22 There are other PCP rifles in this price range, and they will give you the most approachable accuracy in your price range (and a bit more).
Should you purchase the Maximus, and want to improve it, there are several aftermarket components which enhance it's accuracy even more.
You came into the market at a good time, with several different companies introducing sub-200 dollar PCP air rifles. You just have to remember to factor in the price of an air pump, or you will be stuck with a gun which you cannot fire.
Best of luck
Addertooth