i don't know how beginners should do it so i can just outline what worked for me.
i started lessons june of last year and after about 4 months i had my first jam. i was
awful. i could only keep beats and when i started filling i tended to overfill and the concept of time went to the crapper.
my timing only got better as time went on and i had more opportunities to jam with other people. i've already forgotten how everthing i've worked for finally came full circle with regards to my skill and playing ability. i continued learning and practicing linear drumming, fills, rolls, upbeats and downbeats, rudiments, experimenting, and what have you.
all of these was supplemented by the fact that i listen to music alot during the course of the day. lagi pagpapasok ako nakakabit na yung earphones sa tenga ko. pag breaks, etc. i don't consciously try to emulate my favorite drummers but i listen to them so i can get something out of their playing. be it their feel, groove, or some fill. parang... sarili lang siya papasok sayo. pero hindi ko sila kokopyahin talaga. i hope that wasn't too vague.
my advice would be, yes it's ok to emulate your favorite drummers and there's no such thing as the concept of "who you should listen to". listen to anything you want. i didn't improve by listening to dave weckl nor by listening to sean kinney. i improved by practicing, learning, and also using my inspirations as sources of ideas, and not by outright copying them.
i also didn't start out by learning songs. i believe that a beginner should hone his drumming tools and drumming vocabulary first before covering someone's else's work. it's easier that way, for one thing.
pero yun, do what works for you. i only summarized what worked for me, and it may not be applicable to anyone at all.
just keep working for it, and it'll come with time and practice. masarap din yung sasabihan ka ng tao na ok ka tumugtog (hindi rin ako nahihiyang sabihin na isa to sa motivation ko dati para gumaling kasi pangit yung ipokrito ka sa sarili mo na pa-humble effect pa. hehe. pero syempre, number one motivation ko talaga
mahal ko ang pagtatambol. just
keep working at it. you'll get better if you want yourself to. frustrations are normal, you should just find ways to overcome your hurdles. good luck man!)
PS. just so i don't look like someone who talks out of his ass, here's a video of me and my band practicing an easy song. i'm posting it to show that if someone like me, who had absolutely NO talent, could improve, anyone who works hard enough can improve. (hindi ko sinabing magaling ako ha. hehe. bulok lang yan. ilang beses pa ko nagkamali haha.)