I've never dyed my hair before (I think I am the only person I know who hasn't) Anyway I have medium-dark brown hair and now I have a few grey hairs that really seem to stand out against my hair colour.

It's not too bad but I'm guessing now is as good a time as any to dye it. It will have to be done eventually.

I'm just looking for any advice at all as I haven't got a clue....so any brands to go for, any colours that are good, any application or maintenance advice. Any good tips really as I genuinely have zero knowledge on this!

You're always best with colours bought from salon supplies shops. If you insist on buying something from the chemist, then get one lighter than your natural colour!
If you just have a few grey hairs here and there, and you think you can get away with it, I'd suggest you just paint the dye on them as opposed to doing a full head. I'd put off doing the full-head carry on for as long as you can!

My advice is go to a salon.

I thought about buying one of those root touch up things and just dyeing the greys but wasn't sure if that was too weird.

I know this is probably the best way to do it but I don't really like the salon experience. Plus it seems mega expensive for colour.

Nice and Easy is very good. Either that or a trainee night where trainees are supervised. Personally I would go the Nice and Easy hair dye route, the conditioner in the box is to die/dye for

No! No Nice&Easy crap, and no need to spend a fortune at a salon either. Just go to Sally Salon Services or order online - get something decent like Koleston or Majirel that matches your natural colour and 20vol peroxide, mix a little in a bowl, put gloves on, and use your fingers or a dry mascara brush to apply to the few errant greys you have.
You can worry about a salon visit and full-head colour when the greys get really obvious.

I am in same position as OP except my hair is blonde and I have one patch at right temple. I can tuck it behind my ear to hide it but there is going to come a point when I will want to style my hair differently. Would your method work on a streak by temples. How do you get right up to the hairline. How do you avoid irritating skin? I am a hair dye Virgin too.

Yes, you could apply the same method. Using a mascara brush is a great way to apply at the hairline without getting it everywhere, although blonde isn't going to dye your skin like brown will. What shade of blonde are you? In your case it might actually be an idea to start thinking about a few hi-lights.
Incidentally, as a dye virgin, you should always do an allergy test first! Mix a tiny bit of colour, apply it behind your ear, and leave it for 24 hours.

This is the best advice.