Learn Italian: Prendo pure un gelato. - I'll take an ice cream too.

In Italian, it is very common to use the present tense to talk about decisions you are making right now or things happening in the near future. Since you are deciding what to order, the present tense prendo naturally translates to "I'll take" in English.

Why is the word for "I" (io) missing from this sentence?

In Italian, the verb ending usually tells you exactly who is doing the action. The -o ending in prendo specifically means "I take", so adding io is not necessary unless you want to emphasize that you are taking it, rather than someone else.

Does pure mean the same thing as anche?

Yes, both pure and anche mean "also" or "too" and they are often interchangeable. In spoken Italian, pure is very popular and adds a casual tone, much like saying "I'll have an ice cream as well!"

I've learned that "to have" is avere, so why do Italians use prendere (to take) here?

While English speakers usually say "I'll have an ice cream" when ordering, Italians use the verb prendere (to take) for getting food and drinks at a restaurant or cafe. Using avere here would sound like you just want to own the ice cream, rather than eat it!