I'm playing a piece called Spring (by Antonio Vivaldi/arr. Richard Meyer), and the violin solo has this part where you go from F on the E string, to G on the D string. I'm not really experienced with the violin . . so help would be appreciated. Oh, and any tips on not making those squeaky or scratchy sounds ?
Beginner. Vivaldi Spring. The things people arrange these days... At any rate, crossing over a string is done without moving the bow and/or without pressure (depending on piece and speed and technique, sometimes even without contact, namely by letting the bow jump, a feat requiring the kind of balance and coordination making the bow another limb and your bow grip a joint). It's easier to do in the course of a bow direction change, but also possible without it.
As for the squeaky and scratchy sound: try being more economical with your rosin, and make sure it's not just a layer of dry dust, either by having it sufficiently new or by scraping to slightly fresh layers with a knife.
But by far the main asset against squeaking and scratching is practice. Each player is allotted an amount of squeaking and scratching that he has to go through in his life, and the sooner you get the bulk of it behind you, the better.
The other answer is very in-depth and awesome, but I thought I'd add my two cents.
The way I think about string crossings is to take them very slowly. Don't be afraid of slowing the tempo way waaaay down while practicing. Imagine you're crossing a fast moving river on slippery stones that are far away from each other. You don't go racing over them, because even if you don't slip and break your tailbone, you'll get your feet wet. But if you go slowly and carefully, after a while, you'll get more familiar with the terrain and slowly you can speed up.
I suggest setting a metronome for 40-50 and just practice each tough string crossing. Play a quarter note on the E string, then use the next beat to carefully cross to the D string. On the third beat play the D string, and on the fourth carefully cross to the E string. Basically, play on every other beat and keep it slow. After ten perfect tries, move it up 10 BPM or so.
Don't get your feet wet!
Positions are useful here
If you know third position, or even an extended 4th finger, then use it here. F on the E string is just one semitone above E, 4th finger on the A string. By playing that F on the A string you wouldn't have to hop over a string and in third position it's just 3-1.
You can jump the bow from one string to another if you want more attack on the second note, in which case don't be scared of a touch of grind at the beginning of the note. Anything like this just takes a lot of practise practise practise. Practise till you're angry.
Take your bow and press firmly and flat while drawing it very slowly. It makes a really loud awful sound. Now do the opposite and draw the bow as lightly as possible quickly across the string. It makes a fairy, hairy sound. Between those extremes lies the tone you want.
Practise starting from the firm grinding noise (bow flat) and moving the bow faster until the tone becomes clean, and then lighten the touch slowly and turn the bow until it goes hairy. Feel the sensation of the bow against the string.
Most beginner problems with tone come from the bowing action; your arm (and brain) has to learn how to move the bow fluidly and smoothly from one end to the other; it's not easy and it takes time to train yourself to do it. The bow needs to stay roughly in the same place on the string, the pressure needs to remain even, the speed needs to be constant and so on.
The good thing is that you can immediately hear the difference, and if you spend long enough just bowing back and forth and listening to the sound you will get there.
The bow is a paintbrush of powerful magic, and you have to tame it in order to cast a spell.