The companies I know the most about are Rome, Burton, and Never Summer. Hence, I talk about what I know. There are plenty of other good companies out there. Atomic. Gnu. Ride. Stepchild. K2. Etc. I'll leave others with more knowledge to talk about those.
$300 - $350 is going to put you on the "inexpensive" side of decks, assuming you pay MSRP. You want to get an "all-mountain" board based on what kind of riding you said you do / intend to do.
The Rome Solution, an overall inexpensive deck, retails for $390. Other Rome decks in your budget are the Vinyl, Manual (made for people with bigger feet, might be a good idea to look into this one), and the Crail. Rome makes good stuff, and they have great customer service, so don't be put off by these being their "cheap" boards.
Check out the Burton Elite. MSRP $360. It's one of the mid-range All-Mountain decks. I used to to own one. It's a decent ride. The King is wide (for big feet), and can also look into the Clash and the Bullet. You could also check out the Blunt, but that board is a little more geared toward park riding.
They make great decks, but nothing is close to your price range.
In getting all this shit, this is the order you should do it in:
(1) Buy boots, FIRST. Make sure you try them on. Wear them for a bit. Make sure the are comfortable.
(2) Buy bindings next. You need to get bindings that jive with your boots. Sometimes Boot X and Binding Y will not be that compatable. This is why you want to buy boots first. The Arsenls are good, and fit with most boots, from what I know.
(3) With the money you have left over from boots / bindings, THEN start looking for a deck.