for beginner tips; try different play styles until you find the one you like. no one can tell you how to play but yourself, that's the great thing about the elder scrolls and rpgs in general. as for things to do in between quests; whatever you want.
you aren't very likely to be without quests i don't think, you can que quests without limit. i suggest exploring, you'll find random events and locations that should keep you plenty busy.
Don't be afraid to stray off the beaten path. You find a lot of goodies doing that. You can really do whatever you want in these games though. For a beginner I would suggest going through the main quest for a few hours until you get used to the game, and
then maybe join some factions. Or just do some dungeon crawling to level up and get some nice loot.
I always like working on professions in games. Since Skyrim has more in the professions department, I'll definitely be putting some time into that - especially the cooking! I haven't gotten the game yet but I can already see me going around hunting for mats
to use. :)
Start off doing quest, it gives you a point of interest, and somewhere to walk to. You will get distracted and the world will unfold from there. There are a few Shipwrecked boats on the coast that are kinda cool too.