Applying Gang Membership
All the gangs are requiring you to have skill that suitable with them. The Heartbreakers gang would require you to have Social skills, Roses and Guns gang will need you to play Basketball with them, and etc. So before applying make sure you have the skills needed.
Stats to Improve
It depends on your basic stats, but generally speaking, I'd recommend you to increase your Luck first. The reason is that everyone starts with 1 as their basic stats for Luck. Also, there are some certain interaction that would requires your luck, especially when gambling. Each stats upgrade would require you to go to Sick Bay. You can be beaten up to go there, or you can buy some Sick Pills from the Gold Coins gang's shop and go to Sick Bay.
Inside the Sick Bay, go find the Mad Doctor. To increase your luck, you'd require x4 Weather Info and x4 Mirror. Weather Info can be required by Watching TV after your lunch session at Heartbreakers gang's place. It will cost you $10 every time also you are not guaranteed to get Weather Info every time you watch the TV. So if you don't have enough money to spend, I suggest you don't do this.
To earn money without using trait, you need to master Gamble and Banker skills. Then you can do gambling every day after your lunch session. If you have high Intelligence and Agility, despite of gambling honestly, you can cheat your money. This would be riskier but at the same time, the dice roll would roll your intelligence despite your awful Luck.
The other way to gain some money is betting on the Boxing arena. The rule is the same but the reward is different. If you win the money based on what you betting in Gambling arena, in boxing arena there will be some odds involved. So if the odds lower, you won't be winning as many reward as you would earn in the gambling area.
Escape from Prison
To escape from prison, the very first thing that you need to do is to gather your escape team. This escape team should have some skill that you don't have and don't want to learn. So skills like Stealth or Disguise, skills that don't quite useful for surviving, this kind of skills are what you need from your escape team member.
The second thing that you need to do is planning your escape route. For example, Scofield can go to Sick Bay every day. So this Sick Bay can the starting point for him. Bbbat-Man on the other hand is the rich guy, so he can start his escape route from a private cell. You can find your own starting point based on your characters.