Yup, check the easy things, don’t worry about anything you can’t check yourself as there’s no point (and don’t assume that just because you don’t know about it, it’s not a legit problem). Get a recommendation for a local- I’ve got a place I trust, and I will never let them go!
I always do a wee bit of credential establishing if I don’t know the guys, maybe “casually” mention some work I’ve just done myself or similiar, no idea if it really works tbh but I took my dad’s car into the famously crooked garage he insists on using and it’s literally the only time in 10 years they haven’t insisted he needs new wipers 😆
Common suggestion is to find somewhere that doesn’t really do car spannering- a van place, or an MOT specialist. If they won’t work on your car then they’ve got no motivation to “find” some spurious fault. Though on the other hand if there is a minor fault that another garage could fix there and then, they won’t do that either 😉