do they actually save you money?
If you want to save money, terry nappies are an excellent option, as are pocket nappies, but they don't have infinite adjustment for a perfect fit each time the same way a terry does. Also, get or make some reusable wipes - it's no extra washing, no harmful chemicals, no plastic packaging and saves money.
Around 20 small and 15 larger terry nappies, cheap fleece cut into liners, a few nappy nippas and a few sizes of wraps will likely save money on disposables, even if bought new, used for only one child and not sold on at the end (yes, there is a market in pre-loved reusables). Use the smaller terries as boosters with a larger one as an overnight combo for an older baby / toddler. It's an effective, workable, cheap set up.
laundry services, is that worth it? Would that defeat the purpose of trying to save money? I’m not sure how much that would cost and is it a weekly charge?
Not sure, but washing and drying at home is fine. Use bio powder unless you've a reason not to. Terries dry in a single (unfolded) layer so dry quickly, compared to shaped nappies with multiple layers of fabric.
How much nappies would I need for a newborn to start with? And do I need to change sizes as they grow?
IIRC, we started with a few muslins over the newborn days, then 20 x 50cm bamboo terries, and I think we may have used some small prefolds as well, moving onto 60cm cotton terries, and boosting with the smaller ones / prefolds for overnight. You can put a 60cm nappy onto a newborn, but it's bulky!
If you don’t use a laundry service, how often do you do a wash?
This depends on how many nappies you have. If you have more nappies, you can wash less frequently, but will spend more upfront buying the nappies, and need more space to dry the larger load (unless tumbling).
And I wonder if I would be really helping the environment with all the washing anyway?
There are various articles covering this. Check out who has funded the research, and whether they are really assessing 'cradle to grave' impacts when judging the outcomes ;-)