I'd like to play through the entire game once just completing the core missions, then on my second or third playthrough, really explore the side missions. Are there any side missions that "go away" and cannot be done once progress has been made on the main mission?
No, all the side quests can be completed after you beat the main mission. In fact, skipping side quests is a good strategy on playthrough 2 as the quest rewards in playthrough 2.5 will be boosted to the max level of 50, so you can get good rewards such as a level 50 teapot.
You just have to make sure you do not accept the quest until you reach playthrough 2.5.