We recommend NameCheap.com for basic domain registration. They’re reliable, easy to work with & inexpensive.
We recommend DreamHost, BlueHost, or HostGator for basic shared hosting
If you need more power behind your website, try Media Temple. This site is hosted on Media Temple.
A couple good places to get started…
Disclosure. Maroon is an affiliate of the products we recommend (we get paid if you buy from them through our link). That’s not why we recommend them, but a nice perk.