1. Get your room. Check your HQ overview guides how to do that. Leave everything you have in there.
2. Go naked in the core. Start looting shelves and tables with boxes on it (as well as pretty much any container). You're searching for relics. Pick up anything you want in process. Flee from any enemy you see ASAP. You task is to get 20 relics of any kind, you can do it in less than 1 hour if you're lucky. You may need to make couple of returns to put other stuff you want to have. You'll probably find some armor and weapon in the process.
3. Donate 20 relics for the reputation. Now you have 30k to do 2nd level assignments.
4. You need friends since now. It's up to you how to get them. Start doing 2nd level assignments, if you have 4-5 people, you can do most of them, probably except Assassinations and swarm deadly force missions. You can do it with looted equipment from the core on 1st level. Any t2 equipment will do. Lure enemies one by one, that's basically it. That's the way you get new levels and caps in this game.