Lot's of good suggestions here, although overall maybe a bit confusing/overwhelming (no surprise).
Personally I've had good outcomes with the front (small) then back (power) strategy, other than one KT150 based amp as there are no other options besides Tung-Sol. In this case replacing the stock JJ pre-amp tubes with some NOS Telefunken Diamond Mark Red Tips worked magic.
I have not been impressed with GL KT88 tubes vs. JJ. I returned the GLs. This was in my first tube integrated - a LineMagnetic.
A trusted source is important and I have mostly used Brent Jesse at audiotubes.com. Everything he has is listed on his website, and he gives good advice without being pushy or overselling, and has a very reasonable return policy.
Although expensive, if you want to warm things up, the NOS Mullards are pretty special and currently in my tube DAC - however, and this is where Brent Jesse comes in - I have some 1950's GE long plates (branded as Wurlitzer) that I rotate in and out of the DAC that sound remarkably good in the DAC at 1/4 the price.