As many said here, 2-player game is one of the most hardest (if not the hardest). Here some tips for you:
1. First try to win the maps having spots, where zeds come only from one side without welding needed. Then you can cover each other. Try Biotics Lab: if you stay at spawn point, weld door on left and right side before wave starts (in trader time), keep front and front-left doors opened and have a little bit of luck, zeds will come only from the front.
2. Good communication is the key.
3. Don't reload in the same time, especially if both of you are playing the same perk. Zeds can catch you on reload very easy.
4. I don't think support is a good choice before you reach level 3 (when you can carry 2 shotguns). If you don't need in constant welding, I recommend Sharpshooter + Commando combination. Sharpie should concentrate on big specimens (Sirens, Husks, Scrakes, Fleshpounds) and help with small stuff while Commando is reloading.
5. If you need to keep door welded, things are getting much harder. Then I recommend you both choose Support Specialist. One shooting, another - welding. When you need to reload, switch with your teammate and go welding. So you will always have one person with loaded gun to shoot. Both start with pump shotguns, when you have enough money - buy AA12. + backup weapon like Hand Cannon, Katana or M79.
6. Another welding option: one player (Support) is always welding and sometimes provide backup shooting with fast-reaload weapons (Hunting Shotgun + M79 is a good choice), another always shooting: Support with AA12 + HC or maybe Commando with SCAR + Xbow. If welding Support has level 2, he can carry Pump Shotgun + Xbow. So he can weld and also kill Sirens, Husks and Scrakes from a distance. Also you will have a good option to train Scake-killing with 2 Xbow shots, which you will need when advancing to Suicidal/HoE difficulty.
7. Use hand grenades on Fleshpounds. Throw one grenade at his body, fast rage him and throw another 1-2 nades. While FP is doing his rage animation first nade will trigger other nades and make FP blow up or near to death. Be cautious not to blow yourself up. Sirens around can destroy nades with scream, so you should kill them before concentrating on FP. If you are not shooting at FP, you will have 10-15 seconds before his rage. It's enough time to clear threats around him. Just don't panic ))