Hello and welcome to DigitalOcean @lodonnell.
Personally I would recommend you Ubuntu. Ubuntu is very good for new users, who didn't have any previous experience with Linux.
Very important point is number of tutorial that you can find for Ubuntu.
Most tutorials here on DigitalOcean is written for Ubuntu. You can look at them on DigitalOcean Tutorials section. For almost everything you need, you will find tutorial.
I think Fedora almost doesn't have any tutorial here, so it will not be so helpful for new users.
Beside DigitalOcean tutorials, Ubuntu has great Wiki page and very active StackExchange AskUbuntu site for all your questions about it.
Keep in mind, you can also ask all questions here, on DigitalOcean Questions&Answers page.
Anyways I would not recommend Fedora for server, as I don't find it as server distro. It's my personal opinion but I really look on Fedora as Desktop distro and not server one.
For example, Ubuntu Server LTS distros also have much longer support time (5 yrs) comapred to Fedora (18 months as I remember). In that time you will receive new updates all needed.
If you really want to stick with Fedora on server, think about CentOS. Both Fedora and CentOS are built by same company -- RedHat. Main difference is that CentOS is special based for Servers while Fedora are made for Desktop workstations.
About packages you will find on both, I got experience that Ubuntu have newer and more updated packages than Fedora. Maybe it's only me, someone other who used more can comment on it but Ubuntu repositories have newer packages than Fedora which makes using newer packages much easier. Beside that I always find much more repositories for Ubuntu than for Fedora for my every need.
As I said, I really recommend you to try Ubuntu. You will have 0 problem finding anything about it, and for all your server needs, DigitalOcean covered you with everything very well explained.
When you get into Linux, learn more about it, you can try other distros, but usual starting point is Ubuntu.
Good luck, I hope you will not have any problem with it :D