There a two factors to consider when looking at power amps..
1. How much output power do you need (what is your speaker cabs RMS rating)
2. Do you want transparent sound or not (tube amps will add SOME color, SS will not)
#1 - is VERY important. The Rule-Of-Thumb for a power amp driving speakers is that it should produced 1.8 to 2.25x the RMS rating of your cabinets speakers. RMS is also called Program or Continuous power rating - depending in manufacturer.
Thus, a 300w RMS 8-ohm cab should be powered by an amp that can produce between 540w and 675w at 8-Ohms. This will allow the amp enough headroom to reproduce the peak transients that occur in music - without doing physical damage to your speakers.
**WARNING ** - when using a power amp and speaker cab, impedance matching (checking that your amp/speakers both run at same Ohm rating) and gain-staging your system (I put up a long post about gain staging SS amps a while back) when hooking it up, are important parts to ensuring you don't damage speakers or amps.
Also, keep in mind that the "controls" on the front of the amp are NOT "Volume" levels. They are input attenuators. Meaning, they control how much input voltage (level) is required to get the amp to produce it's FULL rated power. If the output level of your Axe-Fx is high enough and the level controls on your amp are almost "off", that amp can still produce FULL power!! Be AWARE that it just takes more input level (voltage) to make it happen than if the knobs on the amp are set wide open. If you have a 500w amp, that 500w can be produced whether the level controls are almost off, or wide open. The delineating factor is the signal strength being fed into the amp.
#2 - while often open to major debate, fact is that a tube amp WILL add some color to your end result, whereas solid state is more transparent. The color may be slight, it maybe a "warming" (or other descriptor) but it will affect your end tone.