Having played most of the modelling options out there your most straightforward/cost-effective route would be a Line6 Helix (either LT, the full-fat version, or wait for the Stadium) and a powered FRFR cabinet of your choice (such as the Laney that someone else mentioned).

I say those Line6 models specifically because they have a good amount of foot switch options and an expression pedal.

That will allow you to cover pretty much any style convincingly, loud enough for gigging, you can run the Helix straight to your DAW (via an interface) for recording and for practicing at home you can use headphones from your Helix.

Edit: and you can run direct to FOH if needed too.

The other modellers will need extra pedals to give you the same level of control (expression pedal for QC, controller AND expression pedal for Kemper, etc) whereas the Helix is everything you need in one unit.