Disclaimer: not a teacher. This is just what my teachers have had me do. Get a teacher.
Violins are hard to learn. You have so much more control over all aspects of sound production on a violin compared to a piano or a guitar (particularly a piano) that you also get a million more ways to do it wrong and make a horrible noise.
As with all instruments, the best way to learn is to have lessons with a competent teacher.
Your specific question about the bow moving up and down the strings - yes that's bad. You want the bow to be perpendicular to the strings, so that it excites just that one spot on the string that you want it to. If it slides during the note, the sound changes through the bow stroke and some of the energy is lost and sometimes it just sounds horrible. This is actually a legitimate technique - once you know how to control it, what it does and you can limit it to happen only when you actually want it to.
So you need to learn to control that. Get your violin and your bow. Stand in front of a mirror in which you can see your entire instrument. Place the bow on the strings, about halfway between the bridge and the end of the fingerboard initially. Look in the mirror. Check the angle of your bow - if it's not parallel to the bridge, move your hand away from your or towards you to change it. This is easier to see if you start in the middle of the bow, rather than at one of the ends.
Now start moving the bow. Does the angle change? Watch in the mirror and adjust. Play open strings only - this is hard enough without having to think about your left hand as well. If you keep hitting multiple strings unintentionally, look down at the contact point on the instrument (not in the mirror this time) and use that to adjust your bow angle. Keep checking in the mirror to see if your bow is still parallel with the bridge.
Go slow, stay relaxed. Don't press the bow down into the strings, you need a little weight from your arm but not very much.
Bow long notes up and down and up and down and up and down. Then do the next string. Then the next.
It will take some time, but this is how I learned to get my bowing angle right on the violin (and before it, the viola da gamba - although I get it wrong on both instruments from time to time as I'm no virtuoso). My teachers both put me in front of a mirror and had me looking and bowing and adjusting and looking and bowing, because they can tell you but there's nothing like seeing it for yourself.