Choosing fuses

[W1RKW]

If one is building a circuit how should one choose the fuse that will attempt to protect the circuit?

I want to fuse protect two circuits in a 813 modulator I'm building.  I need to protect the filament transformer for the pair of 813's and an on board solid state audio amp which will do about 25 watts.

The 813's filaments on the 110VAC side draw around 1amp when stablized after turn.  I put in a 3amp quick blow fuse and it seems to be OK. I see no sagging of the fuse wire when flipping the power switch.

The audio amp draws about 0.2amps no signal and about 1.2amps just before clipping on the 110VAC side.  There's not much of a turn on surge and if there is it is very fast, something my ammeter doesn't respond to.  I figured on a 2 amp quick blow fuse. Again no sagging of the fuse element.

Did I chose right.  When is it OK to use a slow blow fuse vs. a quick blow fuse?

[WA1GFZ]

This will depend on the transformer lighting the 813s. A heater or bulb will draw 10X current at turn on. The transformer will limit this current.
You are corrtet to use the smallest possible fuse that doesn't blow at turn on. Always consider what will happen if the input current becomes 1.9 amps steady state.  A slow blow would allow a smaller fuse but takes more time to pop.