Agree with Magneta's advice.
I didn't take many lessons in the start instead I was learning the easiest songs and progressed to easy songs etc... it was a very sloooow progress and mainly practiced the money beat a million times and some advanced bass drum technique(which probably wasn't the best idea to start with after 2 months)
What kept me going and never stop practicing was that I could see small progress and it was great fun to play along to tracks and actually drum a whole song to the end just after a few months of drumming.
The problem is that it was extremely slow progress and if I would have spent the time practicing rudiments instead I could have learned the songs in a few days instead of months...
Off course you should have some fun as well and try to learn some easy basic rock songs and notice that your rudiment practice actually paid off...
It's a bit like in Karate Kid when the master tells him to wax his cars and paint his house using special movements... the kid doesn't know he learned a lot of karate during the time of this work
That's how I see the rudiments...
Now after almost 3 years I have to practice the rudiments more than anything else, it's basic foundation that I only have bit's and pieces of. (Which make it hard for me every time I learn a new song)
It's not only individual how good you should be after a few months it's also depends
what skills you have been practicing...
For example:
student1: A beginner drummer only learning tunes and play a long to tracks. He could be able to pull of over 10 songs but they would probably be very sloppy fills and messy rolls...
Comparing with student2 that only practice rudiments/fills with a metronome and learning 1 or 2 songs on the side. Chances are that student2 will be able to execute fills in time and clean rolls in those 2 songs.
However student1 have learned other skills student2 haven't. Student1 have got a lot of ideas for fills, beats and listen to a lot of songs how different instruments interact with each other. student1 will probably be much better at jamming with randoms because even though he is sloppy he can predict better when the verse change to chorus, when it's a good time to put in a fill etc...
The point is: Whatever you have learned so far is good practice, focus on your weaknesses at all time and you will be able to do whatever your goal is in the future!
I would say the drummers that stick with student2 approach for at least 10 years have a chance of becoming a professional session-drummer, hired for tours/ drum teachers etc.
Drummers that stick with student1 approach for 10 years either get lucky and join a band that get famous(metallica/white stripes) or just play for fun as a hobby.