Sure, cross bikes can handle trails—up to a point. Depends on the trail and depends on how good of a rider you are. For real twisty, rough, up-and-down trails, not so good; you just aren't positioned right in the saddle, and their slightly long (compared to mountain bikes) wheelbase makes them a bit problematic.
But on what most people consider "trails"? Absolutely. And gravel roads. And, of course cross bikes are very good on roads, especially with the right tires.
Plenty of nice cross bikes out there, at pretty good prices. A lot of bike for the money can be found in Giant's TCX 1 ($1,700). It's a cross bike with a light aluminum frame and a nifty SRAM Rival component group. It has a compact crank for good climbing, you can add fenders to it, do whatever you want.
Fuji also makes a good cross bike at a good price. The Fuji Cross 3.0 also has an aluminum frame, along with a reliable mix of Shimano 105 and Tiagra components. Well worth upgrading as the years go by. Scott's CX Comp bike ($1,350) is very similar.
Most cross bikes come with knobby tires that are sized around 700-32/35. Good for trails and gravel, maybe not quite so good on roads. So you'll need to decide what your emphasis is going to be. For true all-around riding, probably stick with stock tires. But for something that'll perform a little better on the road and okay on many trails, buy a pair of Continental Contact tires in the 700X28 size ($65/pair). At that size, you'll get decent "float" on softer soil. And the inverted tread grips well on grass, dirt, and gravel while rolling nicely on smooth pavement.
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