First of all - yes, you can generate FAST5 files and basecall later. Basecalling during the sequencing run is useful if you want results more quickly. You can also recall your FAST5 files with multiple basecallers, if desired.
There are several ways to basecall currently:
MinKNOW uses an embedded version of Albacore to perform its basecalling. However, the version in MinKNOW occasionally lags behind the released Albacore version.
Guppy is a GPU enhanced basecaller, and has not yet been released by ONT. It is used by default on the GridION. Results are similar, but not identical to Albacore, but Guppy is much faster if you have a supported GPU.
There used to be a cloud version of basecalling in Metrichor, but this is no longer available, and all basecalling must now be performed locally.
I'd suggest looking over Ryan Wick's comparison of the different available base callers, as he has done a comprehensive look (mainly from a qualitative perspective) at the different basecallers.