One of my favorite things to do on this game is to just go around and find new places and enjoy the amazing level of graphical detail (of course it is also fun to just go and plunder random dungeons from time to time).

A few other points....

1) Keep the main quest active for a while. The main quest allows you access to Oblivion which is a great place to level up and obtain valuable and rare loot. On my first play through I ended it too early and missed out on a lot of loot since Oblivion gates disappear after the main quest is concluded.

2) An easy way to train up magic skills is to make a low level spell in any discipline (you have to buy a spell, preferably the cheapest you can find, with the effect you want before you can make one) and then cast it over and over again as you go about your business. Your magica recharges allowing you to train up magic skills fairly quickly this way, (magic levels go up by frequency of use not by the power of the spell) though you need access to the Arcane University in order to make your own spells, so cheap, low cost spells can substitute until you've completed enough of the Mage Guild quests to unlock the Arcane University.

3) On a related point unlock the Arcane University ASAP. Unless you buy the DLC for the game it is the only place to make spells, enchant objects, buy many high level spells, and get a (free) steady supply of many alchemical ingredients among other things. All you have to do is pass a "test" at each local Mage Guild to gain recommendation and you are in.

That is all I can think of now so enjoy. The game is extremely addicting, I have over 180+ hours on just one of my scores, (albeit that includes all the DLC and expansions) so you need never worry about running out of stuff to do.