Q:

How to reduce slow starts at the beginning of the game? 

We recently played a game of Settlers of Catan where for the first 3 rounds none of us got any brick at all, so there was no road construction at all and it dragged a little.  I'm not looking to reduce the randomness, but a slight alteration of the game setup could probably do the trick.  
Here's what I've thought of:

Give players more starting resources.  If everyone began with 2 of everything things could get going faster. 
give everyone an extra free road at the beginning. 

Are there any other good ways to give Catan a bit more of a jump start?  I find if one player gets a slow start in the beginning (like no brick) they can get screwed for most of the game.  Different players being strong in different resources is core to the entire game, but a tiny early lack of wood can make one player suffer through the entire thing. 

A:

Don't forget, you get a resource card for each resource you touch when you place your second settlement. 
If you're still finding things going slow, here's some random ideas you can use:

For each player, remove a Year of Plenty or Road Building card from the deck, shuffle them up, and deal one to each player.   That should get things really going. 
Give each player one extra road as suggested. 
Let each player pick one extra resource when starting. 
NEW - The first time any player rolls a 7 (i.e. this will only happen once), after placing the thief the player has the option of taking a resource from the pile matching the tile they just played the thief on instead of taking a resource from a player (yes, it rewards a little bit of luck, but it also softens the blow of the first thief a little).   
NEW - The first time each player rolls the dice in the game, if they get any resources, they may take one extra of a resource they just received. 

A:

A variant on your, "give players more starting resources" idea:  After each player gets his or her starting resources, let each player take one (or two) resource(s) of his or her choice.   That way, each player can assess what he or she is "short" in and fill that gap (albeit temporarily). 
Another option that you can use with, or instead of, the above: Start with the robber off the board entirely.   The first time a seven is rolled, put the robber in the desert.   From that point on, sevens have their normal effect.   This will give players more of a chance to shrink their hands of resources before getting zapped by the robber, and will keep the robber off the board longer, enabling the towns and cities to produce more resources. 

A:

I agree with warren, I would suggest that faster turns in the first few rounds will solve your problem as each turn is only as slow as the people playing. 
Lack of resources is likely due to one of the following reasons:

Map setup: Make sure the map is based on one of the balanced examples provided.  There are examples of balanced maps available online, use these until you get more experience playing. 
Poor starting position selection: This comes down to experience, but having at least one square of brick in your starting 5/6 tiles is almost a necessity for new players. 
Crappy dice results: The only thing to do about this is just hurry up and take your turn and don't spend ages scanning the board.  

I like to play Catan online as it's a good adaption and here are some things that I found help speed up a game:

Plan your turn when other people are playing theirs and try so you are ready to go when it's your turn.  Try and pay attention to what resources other people have and where they are moving. 
Make the trading section ultra fast.  Don't stare at your cards and think of the 100+ combinations you could possibly make if you just got that one card.  Think about what you need and make your offer.  If people don't have what you need or don't want to swap move on quickly. 
Get to know which numbers come up frequently, so don't place your villages next to brick with 10 or 12, if you can't get 6 or 8 move near to 5 or 9. 

And finally, one common rule is the friendly robber.  Which means you can't place the robber on a tile next to the village of a player with less than 3 points.  This will give players some breathing space, especially those who are struggling to start.