Some excellent advice being given here.
First of all - take a responsible adult with you! I know this sounds daft, but I always take someone with me when I go to a viewing. Most recent horse I was looking for was a horse to compete at a very specific level in a certain discipline, I’m very competent, but I cannot see what I look like on the horse when I’m on the horse - you need a second pair of eyes with a level head. Someone who will help you walk away if the horse is not right, not egg you on to buy something inappropriate.
Your viewing begins as soon as you arrive - you should be able to see the horse being caught, brought in from the field, hooves picked out, groomed, tacked up, etc. all normal behaviour for a horse, but a canny seller can hide an issue by not letting you see that. Once you’ve watched the seller do all of the above (single most important note - always watch the seller do EVERYTHING first, if there is something they don’t want to do, ask why!) Again, watch the seller rider it, ask to see it do anything you want it to be able to do (eg jump, hack, school certain movements etc) before you jump on. Don’t trust a horse to be safe because the seller says it’s safe as houses, you can jump straight on yourself! I know plenty of people who’ve been injured that way. Try the horse yourself in any situations you would want it to be good in - school, hack, in company, alone. Ask to see it load, with clippers, etc if these are things you need to be able to do.
Once you’ve seen the horse doing all the things you want it to, ask to see the passport. Go over this with a fine tooth comb! Check all the markings to see if they match. Weirdly, you can be less picky about the colour - I know grey horses that started out bay, black horses that say liver chestnut in their passport, one of mine says it is roan when in fact the horse is definitely spotted! They all have their whorls, facial markings, hoof colours and such things correct though. I went to view a horse once and the markings were definitely different, even though the pony was black and the passport said it was black. Was not the same pony.
Check ALL the pages of a passport. If it’s been reissued, ask why. If the horse had a name change, again why and what was it called before. Check the date the passport was issued, check the age of the horse and check that the horse actually looks somewhere in that broad category. I went to see a pony that was advertised as 11, but had grey hairs around its eyes and went with a way of going which suggested ‘experienced old timer.’ It’s passport was only recently issued, the owner said the old one had been lost… 🙄
Seriously, assume the seller is there to pull wool over your eyes. Check everything. EVERYTHING! I put out a ‘wanted’ ad once for a small pony gelding. Went to see it, liked it, let my child ride it, groom it etc. put a deposit down and booked a vetting. Vet told me that ‘the gelding op had been successful and what a nice pony!’ We bought him and he was indeed one of the best we’ve ever had, but it was a surprise to find we had in fact viewed a stallion! We never considered checking his little hairy scrotum. 🤣
Ask the seller for references for the activities they say the horse has been doing. If they say it’s done Pony Club - ask which branch and call the DC or chief instructor. If they say it’s hunted, call the hunt master. If they say it’s jumped or done dressage under rules, ask to see its BS or BD record. If they say it ran as a racehorse, ask to see its race record. If it was retired from racing, why? Too slow - ok. Went nuts in company - not so good. Check it all out. Too many sellers say ‘yes, experienced hunter’ but then can’t tell you which hunt it went out with. The number of professional photographers at events these days, there is no excuse not to have a million pictures of it out doing fun rides, fancy dress, or whatever it’s previous owners say it did. If there are no photos at all, or only a couple of grainy photos of an adult riding it when it’s been advertised as a ‘much loved but sadly outgrown pony’ where are all the photos of the child on it? Any parent worth their salt will have taken or bought a gazillion pictures of their darling on the horse.
Ask the seller everything you can think of - don’t be afraid to look stupid, you are going to spend thousands of pounds on something you are going to trust with your life. The suggestions for ‘what does it do if it doesn’t like the look of something?’ type questions, rather than ‘does it spook’ are good advice. Every horse has its limits, even a saint. It’s fair game to check what those limits are. ‘What sort of fences is it more likely to take a swerve at?’ ‘Is there anything it is afraid of? How does it show that fear? A spook? A plant?’
If you like what you see, don’t commit to anything at the viewing. Go and sit in the car with your responsible grown up and ask them what they think. If they think it’s a go-er, go home first of all and have a cup of tea, think about it some more, then call the seller. If you need a second look, should be no problem with a reputable seller. If you want to book a vetting, get a 5 stage vetting with bloods. Ask the seller who their vet is, as they won’t be allowed to vet their own client’s horse. Ask if you can call their vet though, they can only say no.
Main things to avoid imo are buying unseen (what kind of insane person does that?) or not getting a vetting at all.
The only bit of advice above I’d disagree with is checking out the tack. (@OnarealhorseIride says “If you want something quiet, then hesitate if it is being ridden in a kimblewick etc.”) Sometimes there’s a reason a horse is in a particular bit of tack. Ask about the tack. Eg, you’d find my lovely quiet horses are often ridden in 3 ring gags because we do TREC and until recently you couldn’t change tack between phases, we didn’t need a strong bit for the orienteering but wanted slightly better brakes for XC phase so a 3 ring gag allowed us to move the reins up and down the rings as needed. Photos of old tiny pony of ours would have shown him hunting in a Kimblewick because I wasn’t going to send an 8yo child out hunting on a 6yo pony without something more than a snaffle. I had people upset I was selling a hunter in a Myler combination bit with a rawhide noseband. The horse needed something in its mouth (it was 18hh) but responded better to nose pressure than bit pressure (it would try and duck under the bit, the combination let the noseband work first, with the mouthpiece as a back up if he decided he needed to go somewhere in a hurry!) So many buyers had no idea what the bit was they incorrectly assumed it was strong and therefore so was the horse. Fair game to ask though - why is it wearing a flash? A grackle? A martingale? Some sellers have no idea why, it’s just what the horse came with.
Anyway, good luck. I find looking for the next unicorn one of the most stressful things out there. Remember that if you buy the wrong one for you though, as long as it is sound it will be the right horse for someone else, so don’t be too afraid, and don’t think you couldn’t possibly sell a horse that’s not working for you on - both you and the horse will be happier with a better matched partner.