I was just given payday the heist last night and several of my buddies are already in the 30s. Is there a good way I could catch up fast?
If you have a low level, you can play the first level (the bank) in hard mode, solo or coop. The level is easy enough to achieve and you can easily farm the challenges (X frags with weapon Y, etc) by camping nearby the iron gateway or above the safe room.
In general, the easiest way to quickly level up is to achieve the challenges offered by the game. You can find a list of these in the main menu. Some of them will only unlock when you'll have finished others, so you might discover new challenges to accomplish later.
Rob the secret vault (full of gold) in the First World Bank. It might take about an hour to beat the level with the secret vault, but it is worth it. You should get at least 10 level-ups from the secret vault alone. I got about 10 level-ups and I was already at level 100.
There are tons of videos and webpages that explain how to rob the secret vault.
Note: the secret vault can be robbed even in Normal difficulty (after an update). Make sure you follow the steps in the videos precisely.