Budget Tube Amp (newbie)

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.

Old iron IS better than new iron.

Old iron is frequently cheaper than new iron.

Quad II sounds much better than Citation II

I never thought so, but I don’t like Welch’s & ethanol either. IMO, the CII is more coherent from the low mids on down while the QII were somewhat confused in the systems in which I heard them.

In tube sound, the warmth you like is supplied most by a tube preamp.

No it’s not. In all cases, the entire system contributes to the final sound. A nice tube preamp is not going to help a system with a Zippo 250 SS amp and EarBleed speakers wired with HFXtreme cables. A tube amp may be more/less intolerant of cable/LS.

Systems are built with ’recipes’, all little of this, all little of that to achieve a good balance and just like pudding is all in the eating, amps are all in the hearing.