Welcome aboard TF and a potential new member of the Mainship family.
We live our 2008 34HT and have not had any significant issues, at least not expensive ones to fix.
You are likely to find evidence of water ingress (in eng rm) from the hull deck joint. It's common on many boats not only MS.
Checking batty condition would be on my list as that could be a couple of boat bucks if all weak.
I'd be asking for maint records. Has the tranny been serviced, coolant serviced, what has / hasn't been done re heat exchangers (cleaning/ servicing water, oil, after cooler, tranny cooler), gen service, cooler. All engine,ntranny, gen pencil anodes serviced regularly. Have thruster(s) been serviced periodically?
Are all accys, appliances, A/Cs working properly.
Does it reach proper WOT at sea trial w/o any overheating or unreasonable vibration?
Is the ER dry or show evidence of standing water.
My Bacchus website linked in signature has lots of info & projects I've tackled to make it work for us and improve storage & operation.
Good luck