I'm a fan of the Siglent scopes. Which you get depends on your budget. Bieworm's is good at that price point, or if you want to spend a little more, the SDS1104X-E is really good too. That's what I used before I won a nicer one.
There's a TON of videos on using a scope for general use and guitar amp stuff on youtube.
A good / quick info page on how to use it is here:
http://www.r-type.org/articles/art-125.htm
To answer your other question, if we assume for a second that there's definitely nothing wrong with your solder joints, nothing wrong with the wires, or speakers, perfect tubes, and all resistors are correct and in spec, then we would want to look at your coupling capacitors.
Or you might need to chopstick the wires, because a wire being out of place could cause a parasitic oscillation that could destroy your sound entirely.
Chopsticking the wires means moving them carefully with a single wooden chopstick while the amp is on (keep your other arm behind your back or in your pocket). You would need to have some sound input to the amp (looper pedal or tone generator) to be able to hear the difference.
Chopsticking the wires is the easier choice, so start there.
Thanks,
Josh