Thanks for the reminder @Ego! Finally a thread where I feel competent to give advice! :)
If anyone has questions or needs trouble shooting for a sewing project I'm very happy to help (I used to work in a tailors shop when I was a young gal and never stopped sewing all my life).
As for examples of equipment, it really depends on how much you sew and what you sew. If you are making vests/camisoles/stretchy things out of oversized T-shirts you need different equipment than for repairing jeans.
A basic sewing machine will be enough for most jobs, and the less fancy it is, the better for a beginner. My overpriced "Brother" computer sewing machine has been driving me nuts for years because it is such a diva. If it could talk - and I'm SO glad it can't! - it would go like this:
"You pulled on the fabric! How DARE you pulling on the fabric!! I have a sophisticated 7-way-fabric-transport-system and I do NOT countenance fabric pulling by a clumsy human! I will punish you by wantonly destroying my bobbin case which costs 40 € to replace! Ha!").
If you get a used machine it may be sensible to look for a special mechanic to have it regulated. This is money well spent for a beginner because an unregulated machine will give you trouble (jammed thread, uneven stitches, jammed fabric...). Here in Germany it is not expensive either.
There are lots of YouTube videos for repairing/regulating a sewing machine for handy people, though. It depends on your skill level with fixing motors and machines in general, I suppose. Most sewing machines tend to be fussy, maybe due to the strong shaking while sewing? A sturdy table is advisable in any event.
My 35 year old "Pfaff" however seldom gives me a headache (I got it used for my 14th birthday! Still going strong).
The all-time favourite however is a "Juki" industrial sewing machine, a regular workhorse! Only straight stitch but this it does perfectly and without any fuss. There is a strong used market for those, at least in Europe. It sews through leather and layers of jeans without complaint.
Regardless of the machine, using the right needles is crucial for the success of your project. There are sturdy needles for denim, thin ones for delicate fabric, needles with a little blade for leather and needles for elastic fabric. At a certain skill level you can sew most stuff with a medium all purpose needle but in the beginning using the appropriate needle will save you lots of cursing.
There are also sergers, machines which are used to finish off the edges of fabric to avoid fraying. A serger is also great for stretchy fabric, because it can do an elastic stitch (which a regular sewing machine can do as well with some workarounds, but it is more difficult to get a neat result.)
Most important are good scissors, and even more importantly, use this scissors ONLY for fabric. Paper will dull the blades and cutting fabric will be a pain in the ... afterward. Since my kid will steal (and ruin) my scissors all the time I do not buy expensive ones. The cheap ones work just as well as long you are using them on fabric only.
A leftover habit from my tailor shop days is having a steam iron running while sewing. I press all seams directly after sewing them, it makes sewing easier and gives you a neater end product.
This is what I can think of at the top of my head, I'm sure @Sclass has some useful hints as well!