If you do decide to go down the 'buy a domain' route, I'd suggest you not spend more than (say) a tenner for a .co.uk / .org.uk (and as it's a personal / family site, the should be an option you can choose to 'Hide the WHOIS details' ie name and address, so your family privacy is protected).
If you went for a .com / .org / .info then it is possible to get 'anonymous whois' as a service, but many firms charge a fee. name.com makes no charge for that option.
Whether to go for some other firm for hosting depends on what you eventually end up having on your website. If you choose to build pages with a few photos etc (eg places you've visited, rather than a family album!) then most any of the free services should be OK.
Some may plaster pages with ads, or do pop-up ads, but there have been people using free site building setups with no problem, so not all are 'too good to be true' - a few might make their money by offering domains but for higher prices, for example.
slayer: you get a free trial then if you continue to have it free, you have to have adverts. but they tailor the adverts to the type of site, so if it's obvious it's a kids' site, they'll be adverts on child friendly stuff. afaik that's the only snag i've ever found.