Check with local award shop, the businesses that provide trophies and plaques to schools and sports leagues. Many will sell you the blank wood bases used for award plaques outright, and they sometimes have plaques that have gone unsold that you can buy on the cheap.
If you're lucky to have won at a few contests over the years, you can re-purpose the awards as display bases. I've never been one to have a "love me" wall where I display all my winnings (there really aren't that many), so I sometimes remove the aluminum applique from an award, apply rectangular sheet plastic painted to represent flightline, tarmac, or groundwork, and boom! I have a nice base.
Here's an Egyptian T-34/122 conversion on a wood award base, with groundwork created from Celluclay.
Here's a 1/72 F-16A on a base from which I've removed the applique. All it needs is a piece of suitably sized and painted plastic to make it complete. Or a mirror even.
Visit any home furnishing store and you'll find a variety of wood boxes that are used for organizing the contents of kitchen or bathroom drawers. You can paint them (or not), flip them over, and apply a piece of plastic to represent a flightline or groundwork. The Container Store is a particularly good source for these.
Here's a 1/72 Gripen on a narrow organizer (which I have yet to paint and finish) but you can see the general effect that a higher base provides.
When you’re at a home furnishing store with SHMBO, look for the deep, square cubes that are intended to be mounted on a wall for displaying bric-a-brac. I’ve seen them in black, white, and oak. Laid flat with a suitable surface applied they provide a pedestal for displaying models.
Frames can provide the, um, framework, for a variety of eye-catching bases. Michael’s, HomeGoods, and similar shops have a huge variety of bases, so think creatively as you browse the store. Each of the models below rests on a base created from a frame.
If you have a cabinet shop near you, they may have surplus cabinet doors for sale. You can find them in many shapes and sizes, so look for those that will fit certain types or sizes of models. For example, a long and narrow aircraft or vehicle (such as my Su-15 below or an SdKfz 231 8-rad) will look good on a long narrow base, while a shorter aircraft or vehicle (MiG-3 or Pzkpfw Ib) will look good on a square base.
If you need a large base for your next epic, Letterman-esque diorama, I found these at Ikea a couple of years ago. I don't remember exactly how much they cost, but I they were affordable given their size.
Yes, tissue boxes. There are homeowners who wish to hide their tissues (these are the same people who buy knit cozies for their spare toilet paper), so you can find these unfinished boxes at Michael's and other craft stores very inexpensively. Here's one I'm working on, which will ultimately be painted black and a surface applied to better show of this 1/48 A-4 Skyhawk.
Craft store bases
One final option are the unfinished bases you find at most craft stores. Please pardon me if I get on a soapbox, but when I've seen them used to display models at contests, they usually look cheap and shabby and almost always detract from what are otherwise excellent models. Unless you're a master woodworker, I suggest using one of the options I've described here.