I remember reading months ago how fast traveling could potentially mess a few things in the game up. While this may or may not be true, I remember some people claiming that the times in which shops were open were screwed up among some other things. I never fast travel, but I kind of want to start using the horse carriages around Skyrim to take me from place to place. While I know this is not going to cause any major, game-breaking issues, will it cause any minor inconveniences (such as shop times)? And if so, what?

First of all: What? I've never heard of fast travelling altering shop times - it's never affected any shops in my game.
And no, to the best of my knowledge there are no issues associated with carriages. There ARE issues with a mod which gives you real-time carriage travel ... but not with 'vanilla' carriage travel.

I haven't heard of or experienced any problems with fast travel or carriages either.