Firstly does your Mum or Dad have any preferences? E.g. burial Vs cremation, big funeral Vs small, what sort of coffin etc.
Local undertakers should have websites with some sort of price guide and you can email and ask for quotes. You can do this now, if you like (I did).
If you are trying to keep costs down then things like printing your own order of service and booking taxis rather than having funeral limos can be considered. Also, some cemeteries have lower fees for plots for locals.
For burials, how bothered are you about the headstone? I took my time and went to someone a friend recommended for my Mum's stone, spent a good while sorting out the design, wording and stone, and I'm very happy with it - another relative chose a granite/sandblasted stone very quickly and was a bit discombobulated and not too pleased with the result.
What sort of things are "musts", do you know?
A direct cremation or burial and a completely separate wake/memorial gathering is what I did for Mum, after a bad experience with a friend's funeral which overran the 30 minutes slot. Gave us more time to talk about her and a comfy venue with food immediately there for everyone.
Anyway, if you have older friends, colleagues, neighbours, talk to them - they'll have local knowledge and will probably be happy to talk you through stuff.
One thing we didn't realize, our first time, was you're expected to contact a funeral director, well... soonish. We got the death certificate first, turns out that leaving FiL in the hospital morgue for a couple of days wasn't the done thing . We didn't know!