I don't know how far along you are in the game, but the extra experience will help you get ready for the big battles you will have at the end of the first play through. You will also get character skins, weapons (which you may now too far into the game and they could be under powered), and Eridium. At least IMHO, I would say do the side quests.
I do EVERY quest on first playthrough. On second playthrough I just do main story than when I'm max'd level I do all side quests for the level 50 gear. May actually keep a couple characters side quests on 2.5 so I can get the items when they hit the next max level.
"Did I ever tell you what the definition of insanity is?"
I actually do the opposite. Main quest on walkthrough 1 then side quests on walkthrough 2 and then farm for walkthrough 2.5
I do the opposite. Do the main quest and all the side quests in part 1 and by the end you should be somewhere around level 35-36. Second playthrough focus on the main quest because once you defeat the boss all the remaining side quests will scale to whatever level you are. The rewards will also be much better.
When I'm concentrating, I never have more than one drink before dinner, but I do like that one to be large and very strong and very cold and very well-made.
I'm level 20 myself and have been debating what to do too. I think the best thing would be to try and power level until your high enough to do The Chosen and get the Evil Smasher. The game should be much easier after that IMO, regardless of what level or gear your have..
With my most recent character, I've only been doing story missions, and only proceeding with those when I've grinded to a point where gaining another level becomes impractical.
I was level 5 before I fought anything other than Knuckle Dragger. Which was stupid, but at least I got a Hornet. I got to level 12 fighting Boom & Bewm. Currently, I'm working my way toward level 17 at the Drydocks in Three Horns Divide, with The Road to Sanctuary still active.
Boom & Bewm were level 7, and dropped a level 5 Surprise Package. Both a level higher than I've gotten on previous playthroughs, so the grinding seems to have some effect. What I'm hoping for is a higher-level Firehawk. The ones I've gotten in regular playthroughs were level 9-11, which you grow out of entirely too fast.
Going through all those chests in that grinding, I managed to get a Pandemic too, so that's cool.
As a side note, it's easily possible to grind to level 40 against the Warrior in the first playthrough. If you put off doing side missions, and start crossing them off at that point, it's probably possible to start True Vault Hunter mode at level 43 or so.