#1
I've been playing for 2 monthes and I have a couple questions;

#1, I noticed that I'm only stumming downward with my pick, is this a bad habit to get into? Because I see people strumming up and down.

#2, When I try to do vibrato, I tend to tweak up the pitch and it seems like the note is going up in pitch, is it supposed to do this or am I doing something wrong?
#2
1) You should defenately try to strum both up and down, but at this early stage you should be more concerned with getting comfortable with the instrument. Don't worry too much about technique building as opposed to basic fundamentals.

2) A vibrato should be just that - a 'vibrating' note. The pitch will change slightly but overall it should be the intended note. Try searching youtube for 'Vibrato lesson'
-Guitar Gear-
1995 American Fender Strat, EMG 85 pup
Randall RH200 Head
Marshall 1960a Cab
Woods Acoustic
-Bass Gear-
Spector Legend 4 bass
Washburn Bantam bass
Hartke HA2500
Fender Bassman 410H
Play what you love, love what you play
#4
In the beginning alternate picking is very important. Early, bad, habits are hard to shake if you continue with them.
#5
Its important to note that downstrums and upstrums are a little different, especially if you are playing acoustic or very clean electric. While its important to alternate pick, like rbouchard47 said, your use of down strums and upstrums depends on the piece you are playing. What you should be practicing though, is to treat your hand like a metronome and strum up and down to learn and keep rythm, whether you are hitting the strings or not on that up or down motion.