Budget Tube Amp (newbie)

Don't overthink it. Look for something that you can sell for about what you pay. Don't consider anything glittery and new that will drop in value overnight. Regardless of what you hear on these forums, sound quality has not had the quantum jump that some would lead you to believe.
When I had Big advents, many years ago, my Marantz 8B had way more power than necessary. That's a 35 watt a side EL34 unit that's a collectible legend of relatively undistinguished sonics. I've had a couple of those and Model 5's. Great look but way overvalued.

Honestly, my Dynaco ST-35 was a fine match for those speakers. You can find a modestly priced 20wpc unit of some strip pretty easily used. Those that have suggested a mid-priced tube integrated are giving you good advice. Don't jump into the deep end of decreasing returns. A good integrated should give you the essence of what you're looking for.

There’s some good advice already here so I’ll just add this. The comment about good transformers is key, and old iron can’t be beat. Go vintage, but get something with the power supply caps redone unless you can do it yourself. That’s easy with point to point wiring in vintage equipment. Use vintage tubes.
Don’t spend a lot. Start with an integrated to see if you like “the tube sound”. One of the best you could do here is a Scott 299C. It was my introduction to tubes and changed the way I heard music. I had a $$$ surround rig and after hearing the 299C it was game on.. But beware, tube amplifier collecting is addictive. 
Move on to a good tube preamp and power amp. Here I like some of the newer preamps, but stay with vintage power. I have plenty of both but my favs are my Wright 6SN7 based preamp and Quad II power amps. The Wright has open architecture that makes for easy tube rolling, which you will become addicted to. KT66 power amps like the Quad II’s are a good choice, smooth and neutral. A lot of the Dynaco power amps that have been modded also fit the bill. 
Going tube is a journey, deeper than the solid state road. Enjoy.

Hogwash!
Old iron is no better than new iron. A good design is a good design.
A great 60W KT-88 amp is a Citation II, but they are pricy.
Lots of old vintage stuff is in need of a complete rebuild, not just capacitors.
Tube rolling is silly. Get something that sounds well and be done with it.

Just enjoy the music!

Old iron IS better than new iron.

Not that it specs better or anything like that, you just can get old iron for a lot less than new iron and almost without exception transformers are well made. That's the most expensive part of an amp and manufacturers pretty much always made sure that if they were going to tool up a transformer, they did it right. Yes, good design is good design and transformer design has been optimized for years.
For what it's worth, Quad II sounds much better than Citation II.I have three Citation II here right now and that pentode input stage is for the birds. The Citation transformers though are sublime. Properly rebuilt they are a revelation. Bruce Moore for instance gutted many a Citation II with just that in mind.
Yes, tube rolling, just like cable rolling or fuse rolling is basically an exercise in the imaginary side of the hobby...

Lots of good advice here. Try to learn from it. In tube sound, the warmth you like is supplied most by a tube preamp. So you could get a tube preamp  into Solid state amp (there are a lot of them, used available) I’d think of some of the Pass designs like the Adcom 555. I started into the ‘tube’ sound with a used Audible Illusions PreAmp, still made new and with many used ones available in your budget range.