Violinist.com has the most comprehensive string guide I've ever found for classical players. The only thing they have to say about non-classical is to recommend Red Labels for fiddlers, which is terrible advice. Those last forever, but sound terrible.
Johnson Strings has a good summary of what different string types and cores mean. They've also got a very wide variety of strings available at a reasonable price. The other places I've bought strings online (USA) are Southwest strings, and Shar Music. All of these are string shops with a strong online presence.
Shar music has a great graphic showing quite a few different strings, approximate prices, and how they sound relative to each other.
The following is from my experience and readings. All strings mentioned below are strings I've either used myself, or have heard about from multiple people. These are not the only strings worth trying!
The cheapest strings, like Supersensitive's Red Label, or D'Addario brand Prelude are steel core. At the cheap end, the strings I've tried are fuzzy and rough sounding. Red Label was probably my least favorite, but I didn't enjoy the Preludes either. The low end steel strings are also quite popular with beginning students because they are the least expensive, under $20 for a set if you shop by price. Steel core strings have a very long life.
Some of the more expensive steel core strings are have a sharper, edgier sound, like Prim or Pirastro Chromcor. I know a number of American old-time players who use these steel core strings, because the sound they produce is very characteristic of the genre, and the strings are very tolerant of frequent retunings - Old time players will often play outside of the standard GDAE, in tunings like AEAE or GDAD (look up cross-tuning or scordatura for more information).
The steel core string I do use is Helicores, which are a good midpriced string. They have a twisted steel core made up of several strands, and are smoother sounding than the solid core strings like Prim and Chromcor. The sound is still edgier than the synthetic strings I've tried, and than the strings favored by classical players. They are very popular among the non-classical crowd, both amateur and professional. This is what I've been using for the last few years.
There are a wide variety of materials that are used in synthetic core strings, so the types of sound you get varies widely. Some are very good for classical, others cater more towards jazz or rock. I haven't tried very many of them, but I'll mention the ones I know either from experience or by reputation.
Savarez's Corelli Crystal are a synthetic core string. For the money, I haven't found anything that sounds better in classical music, but I haven't tried many other options, either. The biggest reason I don't use them is that I rarely play classical, and the sound of the Helicores is closer to what I want. This is not a particularly widely known string. I did find they needed to be replaced every 3-6 months, while Helicores last me a year.
Dominants are a more expensive option. Among classical players, I hear about these more often than anything else. They have a reputation for both good sound and good durability. They've got a synthetic core.
The people I know who use Evah Pirazzi strings are very satisfied with the sound, but I believe they are the most expensive string on the market. If not the costliest, they are certainly close. The people I know who use those are either very good musicians who will shell out big bucks for the best sound, or amateurs who don't need to worry about money. These strings have a shorter life than the Dominants.
These are used primarily in period Baroque music. They are quieter
than most modern strings, and don't last as long. If you aren't
playing period music or something else that requires gut, you're
probably better off with the modern synthetic or steel strings.
Light, Medium, and Heavy
Some strings come in different weights. Light strings are the quietest, medium in the middle, and heavy the loudest. There are also going to be other subtle differences in how they sound. Most people just use the medium strings, even when the other options are available.
There are dozens more strings that I've never tried, and know nothing about. I'm sure that there are good sounding, long lasting strings among them. But you'll have to experiment to find out what is best for your playing style, and your instrument.
If you've got a limited budget, and are wondering what your options are, call up a violin shop, and ask what strings are in your price range and find opinions online, or simply ask the shop for recommendations within your budget. This won't tell you what the single best string is, but you can at least narrow down your options.
How to tell a violin string needs to be replaced:
If you see wear on a string, it absolutely needs replacing, or you risk it breaking. And while it is unusual, a string breaking has the potential to damage the instrument.
But violinists rarely wait until the string is visibly worn to replace it. Instead, they replace the string because it not longer sounds or feels right. As a string ages, the sound it produces changes. It may ring less or not sound as easily or as good when you bow it. Basically, it is your own experience that tells you when to change the strings. Most casual players will be happy with the sound when changing strings every 6 months or so. Some strings will need to be changed at just a couple months, others will last a year or more. I find that even if I'm barely playing at all, the instrument will sound better with new strings after a year.