and how do you get your 'warm fat tone?'

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Fender Standard Stratocaster
Maton ER90C

I use the tone controls on the guitar,to color the tone,I play alot using the neck pickup only,and middle pup the other half of the time,I turn the amp treble to 6 or 7 depending on the amp...then the bass to 5 or 6....the volume is usually around 5 or 6 at gigging levels.
Turning the bass as high as the treble on the amp doesn't let me drop off much of the treble especially on the neck pup when I'm playing...having a brighter sound with both guitar tone knobs at 10 lets you color it more without becoming boomy...,and turning the volume back always drops some of the highs.
I use vintage Fenders,Brownface and Blackface amps,mostly here lately a '67 Vibrolux Reverb 2x10...oh yeah Weber speakers are in all my amps....the brownface Vibrolux 1x12 is another favorite,a '60 Brown Super 2x10s and a '66 BF Deluxe Reverb...I have others but those are the ones I take out most.
I plug straight in,no pedals or effects...I get the natural tube distortation from the volume.
My Strats all have different pickups in them,Lindy Fralins,Van Zandts,Kleins,and my recent favorite David Allen's.
Want to know about my "sparkly crunchy tone"?...same gear,a little more squeezed pinching tone with the pick and different attack.
At some point in time you get the gear you're fairly happy with,the rest is in your hands and fingers.