If they play Monday and Wed do they go right to the Wed night city after the Monday show? Seems logical in case they run into travel issues.
On off nights other than rest is there anything they typically do such as sporting events, museums , sex etc?
It depends on the artist.
Promotion can eat up a lot of time; calling into radio stations, book signings, podcasts (as guest and in many cases host). Writing and rehearsal, and just extra sleep might be required to put on a good performance. For leisure, they can look up friends that they only get a chance to see when they pass through a particular town. Or they can spend their entire free time trying to find a drug connection in a new town.
It also depends on how big the artist is, and if have people who do the logistics for them.
My friend is the lead musician and doubles as the manager on his tours. He’s exhausted by the end.
If the artist in question is Ray Davies:
I had a friend who had a gig playing for a touring band that was pretty big and impressive in the 1970’s, but had dropped in significance by the 1990’s. Being on the tour bus with the band was among the coolest thing I have ever done. After the Southern Spirits Tour where I first learned I was only a few degrees from fame, they came through town several times and it was always a blast.
The guys in the band loved to go out to bars and clubs that had live music and sort of take over the bandstand. After the first few times when I would get a call once they in town, I would know they were coming and a couple of the guys might come water skiing with us at Canyon Lake during the day - before the show that night. I would usually ride on the bus with them to the venue and then sit backstage while they were on.
If they did leave that night, it was right after the show. If they were not the closing act they would sometimes be rolling to the next gig before the last band was done on stage. (They usually were the headliner, but if not they were on right before the headliner- they were too big to ever be the opening act. They did do a lot of those rock gods of the seventies type shows however. The best was when they did a corporate gig-- no other bands, arrive just before going on, then an hour of signing headshots and right back on the bus and back to the hotel in time to hit the bars.)
When they took off right after the show, they always asked if I wanted to stay on the bus and go to the show in Tucson or LA but I never did that. If they were not leaving that night they would want to go play some more. They did seem to get a high from performing. The best night was at Chars Has The Blues where the drummer and my buddy on keys played with the rest of the locals and closed with a thirty minute version of Brick House– totally rocked the house off the foundations. Another night at The Rhythm Room on Indian School was also a good time. My favorites were the couple of times we went to David D’s Melody Lounge. They didn’t have music every night – but nights that they did were pretty great and the guys liked playing with those musicians, thought very highly of them.
My impression based upon that limited experience was that the days just dragged until it was finally time to gig and then they were finally ALIVE!
I was told from a former roadie (who did not share my view that these were all Southern Gentlemen of great refinement), and my friend the piano player (who was a gentleman from the South of very high refinement) that the schedule was really determined by the crew. They had to set up all day including a sound check, then wait for the show to be over (sometimes waiting for another band to play after our guys were off stage) then strike the entire show, put it into rolling boxes and load it into one or two trucks, strap everything down and sleep on this weird half truck, half bus full of bunk-bed style sleeping births while they drove to the next show. Then they would often set up the next show with local help and only then get to a hotel where they had to double up (the “talent” each got their own rooms). Getting a workout in on any kind of steady schedule was a challenge they always had while on the road.
As a direct answer to the OP, I recall reading that John Lennon and Harry Nilsson would often attend the shows of other performers if they had the night off (and often drew more attention than may have been prudent). In a like manner, The Rat Pack were quite famous for doing the same in Las Vegas. According to many sources, most performers loved having such high profile attenders-- others HATED them for it. Sometimes Martin, Sinatra, and Davis Jr. would get up on stage and crack jokes and sing snippets of songs – which they considered a gift to the artist. The artists didn’t always agree with the high opinion they had of themselves.
It’s hard to avoid drawing attention when you’re John Lennon.
I read that one of the reasons Groucho Marx always kept that greasepaint mustache on stage, was because absolutely no one recognized him without it, and he could go anywhere invisibly.
Well, not if you’re Ian.
Ian Faith : For one thing that goes wrong… one… one single thing that goes wrong, a hundred things go right. Do you know what I spend my time doing? I sleep two or three hours a night. There’s no sex and drugs for Ian, David. Do you know what I do? I find lost luggage. I locate mandolin strings in the middle of Austin. You know? I prise the rent out of the local Hebrews. That’s what I do!
Indeed. Canadian blues artist Anthony Gomes was stuck in Pittsburgh after a show Thanksgiving Eve. I ended up celebrating Thanksgiving that year with Gomes, his band mates, and a bunch of other people without Thanksgiving plans. Possibly my best Thanksgiving ever.
In the 80’s, the “News” from Huey Lewis and the News, would come into our pawnshop looking for bargains. They said it was something that Huey had taught them to do before he made it big.
Were they forced to turned the bargains over to Huey? Be kinda funny if it was indentured servitude deal.
Say the wrong thing and you’re old News.
We just pass the time in the hotel rooms
And wander 'round backstage
'Til those lights come up, and we hear that crowd
And we remember why we came
They were workin’ for a livin’.
If the artist is Pete Townsend:
Sheraton Gibson Pete Townshend Sheraton Gibson - Google Search
He screwed that up because the Sheraton Gibson was in Cincy, not Cleveland. 
The ability to FaceTime with families back home, and to play intense video games from the tour buses, has really positively changed the touring experience for many musicians.
Much healthier distractions for pumped up players than back when all they could do was drugs and get into trouble, after the gig.
If you’re Steve Miller, you walk around Milwaukee with the rest of the band.
He played one of the free stages at Summerfest on a Friday night, and the next night Ann Wilson of Heart cancelled (Covid). Miller called up and said “Hey, me and the guys are just wandering around town. We could drop by and do another set.” (And his energy level was great that second night, at the age of 78!)
What is it with Milwaukee? Jack White was just photographed at a Brewers game, and I just saw a great photo of Mick Jagger at a Milwaukee bar where no one recognized him.
Or, per lawyer/car enthusiast Steve Lehto, you go outside of town (with the tour bus) to see a one of a kind Ferrari. (He seems to recall that it was the J. Geils Band.)
This is an artist:
I don’t know if they tour much. When they do, I suppose they paint.