The bulk of combat (in my experience) is spent letting players make decisions. Monsters have a relatively simple "sheet" (barring Solo monsters), so they're easy to run. The problem is that players are often overwhelmed by their choices. I would recommend running the game straight until you're comfortable with its benefits/flaws, but here are some options (note: talk to your players beforehand about these, so they're on the same page), and don't try all of these at once:

DM Side:

1) Halve monster HP, double monster damage. This will ensure that fights are over quickly. This makes combat a lot swingier, but the players should be able to handle it nonetheless. Doubling minion damage makes them slightly more threatening too!

2) End combat once it's assured that the PCs have "won". You'll get to situations where there are one or two non-elite monsters left who are bloodied. Or maybe the PC hits the monster and brings him to 3 hit points. Just end combat earlier! The monsters surrender, or you declare them "dead" despite having a handful of hps left.

3) Standardize Initiative. Have everyone sit in accordance to their initiative "bonus", with the players with the highest initiative sitting to the left of the DM. When combat starts, that player and the DM roll off to see who goes first, turns proceeding in clockwise order. This cuts down on "whose turn is it?" a bit.

Player side:

1) Avoid situational math. There is a lot of fiddly bits in regards to 4e math; players can pick up feats and magical items that give them bonuses in very specific situations (i.e. "Add your Con bonus if you knock the target prone", "+1 to defenses if you teleport more than 3 squares" etc.). These bonuses can slow the game down, either as players try and add them up, or as players remember after the fact that a bonus should have applied. Encourage your players to take feats and make items that offer a relatively static bonus that they can account for easily.

2) Have your power picked before your turn comes up. It's easy to sit back and watch other people toss dice, but that's a lot of time you could use to be ready to act when your turn comes up again. Granted, things change, but for every time some monsters burst in from behind and flank you, there will be other times where you successfully predict what will happen. In those times, you'll be ready to roll some dice.

Play an Essentials character. The Essential's characters are (generally) a lot less complicated than their original counterparts. Especially if a player is new, I would recommend trying an "essentials" version first, and if they want something more complicated, let them switch to a "non-essentials" version if one is available.