After the initial meld, can I do multiple moves in the same turn?
You can do as much legal "moves" as you like in the same turn. After making the initial meld, rest of this player's turn is treated just like any other turn.

Things you can do in your turn (multiple times):


add tile to existing group/run
put set of tiles creating new group/run
break existing set (group/run) into separate tiles, as long as all of them will be left on the table at the end of your turn, and each one will be part of some legal set.
obtain Joker, by putting the tile it's replacing in the Joker's place on the table. Remember that you can't take any tiles from the table, so the Joker has to be used in some way before you finish your turn!


I know you said you're not interested in playing the game online, but this seems like the easiest way to grasp this. I feel that explanation like this is much more confusing.

If you want, please take a look at official rules.