Do you DRIVE or “motor” (to get somhre)?
Is 'motoring" still in common use?
You drive. “Motoring”, as a noun, is used, although it sounds a little old-fashioned. Or something can be “motoring along”, meaning moving quickly.
“Motoring” is a pasttime, not a verb. Top Gear is a motoring show. But you don’t say “I’m motoring to the supermarket”, but you would say “I enjoy motoring” if you were into cars.
ETA: Also, what Usram said.
Irish doper here. No one I’ve ever heard would ever say “I motor”, it would always be “I drive”. The only way I’ve heard the verb to motor in in the saying “I better motor on”. One might say this regardless of whether they were driving or not.
I think the only situation I’d use ‘motor’ as a verb would be to say I was ‘really motoring’, and this might not be related to driving, but with getting through a lot of work quickly, or whatever.
What the others said, but also, it will often just be go - as in: “Let’s go to the beach tomorrow” - Driving being implicit where distance dictates it.
Saying “Let’s drive to the beach tomorrow” might seem awkward or redundant, or might almost seem to imply that the driving is the end, rather than the means.
“Motoring” sounds vaguely edwardian; i like the word (as a verb). Pray tell, is “automobiling” still in use in old Blighty? Would one go “automobiling” to get to “Luncheon”? 
‘Still’ in use? Was it ever?
No. One would not.
One could go for a jaunt before taking tiffin, though.
Pity, I fear I shan’t (I always like "shan’t); that’s a cool contraction!
I’ve been known to use shan’t, dunno where I picked it up but people often comment on it so it can’t be in that common usage (but I’m also the guy who says amn’t!)
I definitely use ‘shan’t’ in speech
I defiantly use “shan’t” on occasion.
I swear, nobody has said “Jolly nice day!”, or “Top hole, sir!” for at least 60 years though. 
Austin Rover used to have an advertising slogan, ‘Now We’re Motoring’ [you mean we weren’t before?] in the 1980s. That’s about the only context in which you might hear it - and even then it might be as a metaphor for some activity that didn’t involve cars at all.
“On Boxing Day, Hugh and Cissie are motoring up from the Wallops” is another fragment of retro-style advertising copy that comes to mind from some years ago. I can’t think of anyone who would actually use it today - just sounds very old-fashioned.