Know what you want before you start to look , sit down with a cup of coffee ( and possibly your instructor ) and write yourself a list of essentials that need to be asked - height , age , type , build , sex , ability (of rider standard ) horses experience and standard of schooling, allergies , vices, faults and issues . How he is kept ( grass / stabled) is he good alone or in company ? Open to vet , good to catch , shoe, clip, handle, traffic, load ( trailer / horse box) turned out alone / with others etc etc does he come with tack and / rugs ( although these might not necessarily fit properly just because they are included)
It probably sounds like I am being unfair but the fact you have questioned this in the first place puts you into a pretty inexperienced bracket so I really would get help from someone who is more experienced it your instructor or yard owner .
Most of your "weeding out of unsuitable ones " can be done looking at the adverts , ring up only those that sound like they really are suitable .
Make yourself an easy to use check list and whilst chatting to owners on the phone tick off as each point is covered then ask about anything they haven't covered that is important
Money - buying a horse has a lot of unseen expense , set your budget and work backwards to how much you can afford to spend on the actual horse after you have covered the other expenses ( veterinary 5 stage pre purchase exam , transport home , tack , rugs, insurance , up front livery fees etc etc )
If you do your homework thoroughly you will not be a time waster , don't get pressured into being sold something unsuitable , you must feel happy and safe with your new horse . You don't need to buy the first one you see but on the other hand if he's perfect don't miss out but go back and try it again just to be sure ( taking your instructor preferably )
Good luck and keep us updated 