wax will seal and give the paintwork some protection.... it's pretty easy to do yourself, the key thing is the understand your product.
most will be better used with a very thin layer, leave and buff off with a cloth once 'cured', then put a second one on for best results.
some waxes you can go around the whole car before they cure, some cure almost straight away... so deciding on the wax and understanding how it works is key.
for someone new to this sort of thing, I usually say go with a wax like FinishKare 1000P Hi-Temp Sealant (don't let the sealant in the name fool you, it's a wax paste)
the reason I like this product is that it's cheap, you get a massive tin, it lasts for ages on the car, and you can wax the paint, wheels, exhausts with it.. so very good value... just put it on very, very thinly.
as for a car wash place cleaning the underside - I wouldn't risk it.
**caution - sweeping generalisation coming up** mostly all road side 'quick' car washes damage paint and cars with the way they wash them... and I would be worried that they would just stick their high powered jet wash underneath and do more damage than good.
but then, that's just me... I am a bit mental, with OCD and a long time detailer..... :lol: