How do you read key signatures on the violin?
Key signatures are the same for violin as for any other instrument. There are many good resources on the web that detail the meaning, history, and use of musical key signatures. To get you started . . .
If you're asking, "How do I know where to put my fingers, based on the key signature?" Well, that's a common question for many students. In order to simplify things for beginners, most method books start students with a straight fingering - high first fingers, high second fingers, and low third fingers. This can be rather confusing, though, when you start to get into sharps and flats.
I find that it's simplest to think about it like this . . . On the piano, the sharps and flats are clearly visible. They have black keys. The white keys are the natural notes - notes that are not sharped or flatted. On the violin, we don't have the luxury of different colored keys, so we simply have to memorize where the natural notes are played. In the first position, that would be . . .
For example . . . The 2nd finger on the A string is a C. If played in the high position (like students usually learn) it is, in fact, a C Sharp. And, if you look at the key signature in your old beginners method book, you'll see a C sharp in the key signature. To play C natural on the A string, you would do so by placing the second finger in the low position, beside the first finger.
You just need to memorize where all the natural notes are, and then adjust the fingers accordingly for sharps and flats.
Without getting further into theory about key signatures, I'll just point out that it is easier than it sounds, since the sharps and flats of the key signatures always come in the same order. If the key signature has one sharp, it will always be an F sharp - raise the 2nd finger on D and the 1st finger on the E string. (That would be the key of G Major, by the way.) With two sharps, you'll still have the first F sharp, but now you'll also have a C sharp - raise the 2nd finger on the A string and the 3rd finger on the G string.
I recommend "Scales of the First Position for Violin" by Harvey S. Whistler, if you wish to learn the commonly played keys and their fingerings. But, at the very least, memorize the naturals! Best of luck to you!