Things to look for are, as you might guess, bobbins that fit the machine (several machines I have bought had bobbins that came from an incompatible machine), a bobbin shuttle and carrier (for transverse and vibrating shuttle) or bobbin case (for vertical round bobbin machines like singer 15 or White rotary) and that all parts of the irons are present. I have seen machines here that were missing the metal foot pedal, large metal treadle wheel and the pitman rod that connects those two. Push on the foot pedal with hand or foot, if the large treadle wheel rotates then you have the needed parts. Don't worry about the leather treadle belt, they are easy and cheap to replace.
Personally, for a first treadle I would look for a Singer 66 or a White Family Rotary because they are most common, cheapest, and they take the same needles (15x1) that you would use in today's home sewing machine. I don't want to deal with machines that use odd sized needles, partly because I don't know what damage might have been done if someone else used the wrong needle.
The cabinet condition is most important to me because you can swap out the head later if you find a better one. Of my three treadles, two have warping or separation of the wood veneer on the top. I don't care much because they were free or cheap and still functional (after a light sanding and soaking in linseed oil), which is what is important to me. The third ($55 off eBay) is so beautiful, only missing one of the pressed wood carvings on the front of the drawer. It is a Standard Rotary, which contradicts what I said earlier about singers and whites as a first machine. It take regular sized needles, which is good because I didn't know what to look for then!