There is lots of good advice in this thread already, but I'll just jump in on a few points based on my personal experience (YMMV):
Don't bother with a recruiter at this point in your career. A studio will only pay the additional recruiter fee when they're having trouble finding a certain kind of candidate, such as for a senior role or something very specialized.
Expecting to land your first job in 2-3 months is going to be a challenge. It took me about two years to land my first development job. During that time, I worked in customer service and QA for a game publisher during the day, and I'd go home and write code for a few hours every night. Basically, all of the time I used to spend playing games, I would spend coding instead.
Spend all of your free time coding and learning new things. You can also learn by volunteering with an indie or mod team. You will learn a lot by seeing how more experienced people write code. This is why having access to all of the UE4 code is invaluable, and there have to be some great learning opportunities on GitHub.
When you're trying to get your first job, apply everywhere. If you don't want to work there but get an interview, that's good experience. If you don't see a position open, apply anyway. I got my first two jobs applying to places that didn't have a position open that I fit, but that were opening new positions soon and I basically got in at the head of the queue.
Networking events are cool, but I don't think I've every hired somebody that I've met at one of those mixers. It's definitely worth your time to go, but go forward with the assumption that you need to chase down opportunities through other channels.
Be persistent, and don't give up. Landing your first gig is really hard, but it's worth it!