Say for example team X going from a 3 point fav to a one point dog, or basketball team opening the day giving 2 but by game time giving 7. Who do you go with in these situations? Are they really bad mistakes by the line makers or is the public just loading up on cetrtain teams?

The linesmaker does make mistakes, but people also perceive value and rate teams differently.

The "public" usually doesn't bet enough during a football game week to move lines, except on game day. Line moves during a week are usually caused by certain large bettors moving a (comparatively) large amount of money on one game. Some people call these moves "wise" or "sharp", but following them blindly is a recipe for losing. Fading them blindly would also be a recipe for losing. Some win, some lose. Big moves that start late in the week and continue through game day often have significant public involvement. There are systems to figure out the general nature of a move...i.e. is it a bunch of "public" bettors or a few "wise guys" moving a line. sportsinsights.com is a good place to start.

Football is very different than basketball though. There is a lot more money on both sides of every football game. It takes a lot of action to move numbers.

Hoops has a much smaller daily market. The liquidity is much lower. Lines move with much smaller $$ on one side or the other, and some books even consider plays of proven winners in moving lines, rather than the volume they might be playing. Sometimes you will see hoops lines move one direction even though a large majority of the "public" might be betting the other side. Clearly the books took a position in those cases based on what other, specific bettors were playing. There are "smart" money strategies in hoops and there are ways to identify who might be making the moves. If you're tied in with one of these systems, you can do well. Following or fading blindly though usually doesn't work.