I successfully completed my encounter. I was allowed to move one space for my success. If I move to a space with a monster can I then have a combat with that monster, or do I have to wait until next turn?

Did the resolved encounter instruct you to move and have an encounter, or just to move? If the latter, you don't have to deal with monsters until the next turn's encounter phase.

As general rule of the thumb, remember that during each encounter phase you're allowed to resolve only one encounter (unless there are monsters on your space and you kille them all)

When you have an encounter with a monster, does the encounter continue during thAt turn until the monster dies, or your character dies...or do you do just one roll for sanity and combat that turn and then you have to wait till your next turn to combat again?

You do one will and one strength check for each monster. If you kill them all, you can have an encounter with something else (generic, gate, clue, etc.).

When you have an encounter with a monster, does the encounter continue during thAt turn until the monster dies, or your character dies...or do you do just one roll for sanity and combat that turn and then you have to wait till your next turn to combat again?

You only fight each monster once during each turn. If you want to fight the monster again, you'll have to wait until the next turn.