What do I need to consider for a cheap wheel for a Specialized Tri-Cross to use on a trainer? I am wanting to buy a new rear wheel to make my life easier when wanting to use my turbo trainer, so I don't need to change the tyre on my current wheel every time I wish to go out/stay in.

My current wheel is an AlexRims Ace19 with a internal width of 17mm.

As this is my first time buying wheels, what would be some considerations for getting something cheap that fits the bill? I say cheap because I'm using a used adventure bike, so nothing is expensive in my setup.
From your other question we know you have a Specialized Tri-Cross 3x9 with an XT MTB derailleur.

This is what you need to know:


Rim brakes and quick release hub. 
Rim diameter: ISO 622 or '700c' - you can look up bicycle wheel rim diameter conventions
Rim internal width: You can go a little narrower or wider than 17mm as tires work on a range of rim widths. Check the min and max rim widths of the trainer tire you will be getting.
Hub width (between the frame dropouts); Your bike will either be 130 or 135mm. It's easy to measure with the rear wheel out.
8/9/10 speed compatible. The cassette freehub body width for all those speeds is the same. On 11 speed wheels it's a little wider to accommodate 1 extra sprocket. 


You don't care about spoke count, weight or stiffness so you can literally get any wheel that meets those specs. I'd look around on Craigslist or Facebook for a used wheel - but make sure it's reasonably true and the hub bearings are not worn out. Alternately you can get a new cheap wheel from either your LBS or one of the online bike component stores.