With the new patch I'm finally going to start a new game of Skyrim. However, I know quest bugs still exists. With that in mind, are there any to still stay clear of? Also, are there certain quests that should be done before others to avoid any bugs? (I plan on joining the Storm Cloaks btw) Thanks!

One bug which I don't think has been fixed (judging by the patch notes) is that during the civil war quest (for either side), when you are sent to one of the camps to get orders from one of the generals, then depending upon how you 'approach' the tent, then sometimes dialogue doesn't appear properly, such that you are not able to get orders from them. The best advice is, to manually save at the start of each part of a quest (keeping a cycle of about 10-15 savegames) and don't start overwriting savegames until your current quest is complete (this applies to 'all' quests). This way, if something does break, you can go back to an earlier save and try again. Frequently, it will work properly, unless the script is fundamentally bugged, in which case there is no way around it. However, using this technique, I have managed to do much of the game without encountering a quest that becomes stuck. I encounter bugs all the time, but keeping several savegames and trying again will usually allow you to get past the problem. In my experience, the game is more 'twitchy' than fundamentally broken.