Yep, the XLR cable out of the DI should run into your mic input. As Farview says, you can run the bass into the line input without a DI, but you may not be happy with the result. And it's true that the DI converts from line level to mic level (hence the need to output into the mic pre), and instrument level is roughly equivalent to line level (close enough), but the major difference is that electric guitars and basses operate at much higher levels of impedance. The DI also steps down the impedance to what the mic pres are looking for. (This is why some of those standalone recorder-mixers will have dedicated "Hi-Z" (high-impedance) inputs for electric guitars and basses. (Even some mixers have these now.) Note that things are somewhat different if your bass has active pickups that require a battery.
One other thing: because you run an XLR cable between the DI box and the board, the DI box can be powered with the board's phantom power. There are both active and passive DI boxes, so for active ones, this is handy (no batteries to replace).
And if you're interested in particular brands, I've always been happy with Radial's boxes, though they make some pretty pricey ones. I think their entry-level one is $150 or so. And Behringer (yes, Behringer!) makes actually a pretty tough and reliable little unit, for considerably less. At the higher end, DI boxes will have amp simulation or other tonal alteration possibilities.
Joe R.