I'm not an expert on the Apocalypse World style games by any means, so take anything I suggest with a grain of salt. :)
It sounds like the combat you described went fine up until this point:
6. Wizard casts MM, rolls a 7. Chooses to draw attention to himself. This is where is gets screwy (for me). In response, three of the goblins charge the wizard. One is knocked off his feet by the Fighter's defend move, the other two beat the wizard with their clubs. Was I supposed to roll anything here? Do the goblins just automatically hit, and automatically do X damage?
Having the goblins automatically hit the Wizard and deal damage seems like a Hard Move to me. I would have done that if the Wizard player rolled a 6 on his Cast A Spell attempt, as a miss means the GM can make as hard of a move as they want. A 7-9 is still a partial success.
Instead of having the goblins hit and deal damage, maybe a better GM move would have been to put them right up in the Wizards face. Have the goblins identify the Wizard as the biggest threat (attracting unwelcome attention) and start to rush towards the character (putting him in a spot). This way you are setting up your next GM move. "The goblins shriek and rush towards you full tilt. They'll be on you in a slipt second. What do you do?" If the Wizard just stands there or gets a miss on the dice,
then I would have the goblins beat him silly.
7. The Fighter and Cleric start hitting goblins, and the Wizard starts trying to cast invisibility on himself (was never succussful, ends up dying lol). I don't have turns as such for the goblins, they just get to hit the players whenever they roll <10 basically.
Generally, the GM only makes a Move when the players miss a roll (<7) or ignore a threat. Remember, the 7-9 result is still a partial success. The Hack & Slash Move specifically says that the character takes monster damage on the result of a 7-9, but I think that's just about the only Move that results in damage on a "soft hit".
8. At this point, the thief has been sitting patiently, waiting for his 'turn'. I had to break up the flow of the combat with the other guys to deal with him.
I think this is just a skill that comes with practice. Some players will wait until they're asked for their action while others will jump right in. As the GM you just kind of have to be aware of how much time you're spending with each of the players and try to balance it. Sometimes it's easy to do that, as each Move can sort of be one turn. "OK. Wizard, you tried to cast a spell, Fighter and Cleric you both traded swings with the goblins. Thief? What are you doing?" Sometimes it's not that easy though. Generally I try to switch back and forth between groups when it's both a) logical to do so (meaning that we've resolved everything related to a single Move) and b) when it's dramatically appropriate. Cutting away right before something resolves is a great way to build tension.
I'm also unsure when I'm supposed to use any of the GM moves. When there's a lull? Cause with my group, someone is always wanting to try something.
Again, basically you're supposed to make a GM Move when your players look at you to find out what happens. When that happens, you make a soft move and ask them "What do you do?" You also make a GM Move when your players miss. If your group is always trying
something, then that's great because it means that they're making Moves and should be rolling dice. Some of those actions will succeed outright (10+), some will snowball into more Moves (7-9), and others will fail (<7). When they fail, that when you make your Moves.
If you and your group are having fun, then you're not "doing it wrong". MCing or GMing an
Apocalypse World-style game does take some getting used to. Listening to
The Walking Eye Podcast's
Dungeon World AP episodes really helped me. Maybe they'll help you too. :)