For anyone interested in learning how to make your own silver jewellery, this blog has my recommendations for your first tools. It's very easy to get carried away thinking you need everything but with a few basics you'll be well on your way!
What you want to make is a big factor to think about it. If you are just starting, using a torch might be a bit daunting but you could always ease yourself in with some wire wrapping, using components with loops or jump rings already attached or drilling a hole to thread through. Then you can build your tool collection as you get more confident!
Basic Hand Tools
The things I use most often would be my pliers (Flat and Chain nosed, Round nosed) files, rawhide mallet, my saw frame, blades and bench peg and some sandpaper in a few different grades. This allows me to saw out shapes, open and close jump rings, make loops and twist metal and more. These don't need to be wildly expensive either. My bench peg was literally a piece of scrap wood and a G clamp from B&Q for YEARS. In fact I still use it now.
Nice things to have would be parallel pliers, some polishing papers, beeswax, a steel ruler and a scribe to mark designs on metal and something to add texture with. My favourite texturing hammer is actually one from the hardware shop down the road, I rarely use my expensive actual texture hammer so don't think you can't use what you might already have.
Soldering
If you are wanting to join metal, create solid rings and bangles, layered pieces and stud earrings you might then want to start thinking about a soldering kit. You'll need a torch and some gas, a soldering block, some kind of flux (Borax or Auflux), paintbrush for the flux, stainless steel tweezers, reverse action tweezers, silver solder (hard) and some safety pickle in a container and plastic tweezers.
If you want to get fancy, you could add another soldering block (useful when heating larger items as you can bounce heat back off it), a charcoal block, a soldering probe for pesky bits of solder that have moved, medium and easy solders for multiple joins, a third hand and a plastic mesh sieve to contain little bits of metal rather than trying to fish them out with the tweezers!
It's also good to have a larger soldering mat to help protect your worksurface from heat and any stray bits of hot metal or you could use a baking tray as a container.
You can buy these items in kit form from places like cooksongold which can make it easier if you want to be certain you are off to a strong start.