Sewing, Repairing and Reducing Waste

Singer are by far the best for finding replacement parts and repair guides, followed somewhat by Brother. I would recommend avoiding the others if you want to be able to maintain or repair the machine in the future.

The stitch patterns should be documented in the manual. Otherwise they are somewhat self explanatory. The symbol looks somewhat like how the stitch will look. Straight stitch is usually a straight line (possibly dashed or dotted), and is usually the first pattern listed. Zigzag stitch looks like a line that moves from side to side in evenly-spaced triangles and is usually the second pattern.

There may also be various asymmetrical variations of the zigzag stitch, or ones that get narrower and wider over time, these are probably what you're referring to as "all the other symbols". Most of these will function as stretch stitches but you don't need to use or care about them unless you specifically want them for their appearance. They make no functional difference. Some of the more elaborate ones may not stretch evenly or adequately.

If the machine has a straight stretch stitch option (as I explained in my other comment) then the symbol for this may vary so check the manual. It's usually some sort of a straight line with dots or dashes.

Some machines also have a button hole mode which is usually listed alongside the other stitch options. Again the symbol for this can vary but it is usually some sort of a rectangle with wavy lines, and it is usually clearly marked in a different color or with a border around it or similar to distinguish it from the others.

Regarding needles, spools, etc.: Needles are completely standard and replaceable and interchangeable on all machines except antiques and this should be the first thing that you do anyway. You can get a twin needle for specific tasks but generally you won't need one, it is not "more useful" than having just a single needle (this is not a "two is better than one" situation). The most important thing to make sure that you have is the presser foot, there should be one attached to the machine and most machines are intended to include one or two other types that can be swapped out (e.g. a narrower one that is useful for particular types of fabric or for working in tight situations) and if these are missing then you will have to try to find replacements or do without as they are mostly NOT interchangeable between machines of different brands or families. Check the manual for what accessories should be included, and make sure to look for them in the accessories compartment. It would be useful if the machine included one or two bobbins so that you can get started sooner but these are mostly standard and easy to find online (there are a few different types so make sure you work out what type you need).

I would highly recommend watching a few beginner/introductory machine sewing tutorials so that you can learn about the various parts of the machine and how it is set up and used, even if you aren't planning on doing any actual sewing from scratch. This will make it a lot easier to know what you need and understand what you're looking at.