If I did create an index called "website" as in the documentation and stored all requests for my website in there, that index would grow forever. What are some tips for keeping the index size down? Is it common to name an index website-08-04-2016?
Yes, that is very common.
Also common, and a good practice, is to provide aliases for your indices to abstract that away from some users, so, for example, you can have an alias called "website" that includes a list of those indices with date suffixes. When you add or delete indices, you can update your alias, and users/application software that uses "website" for searches will be undisturbed.
It's a bit more work, but you can follow a similar pattern by size. Monitor the size of the index to which you are currently writing, (maybe it's called "website-part01") and when it gets to the size you want to limit, create the next one ("website-part02") and update your alias. You can also have a different alias for indexing. It can only refer to one index at a time, and you can't alias an index that doesn't yet exist, so you would manually create the index, and then issue a command to atomically remove part01 and add part02.
Something to be cautious about with this approach is that you can end up with a lot of shards after a while, and that can seriously effect performance. In most recent versions, though, you can reindex in ES, which gives you a way to change the sharding of an index, or combine multiple indices into one.