I was in the same position very recently that you are in now. People who work on tube amps and such tend to recommend getting an analog scope, partly because they can be had for very little money or sometimes free and partly because analog scopes have a smooth and responsive display that is nice for audio. Dave (the owner of this site) recommended to beginners to get a analog scope to start with, because they can be had for little money and because they are simpler to use as a beginner, having all physical controls, without alot of advanced functions getting in the way. And the thing with some of the old analog test gear is that it was built really well, to be repairable, along with service manuals including schematics being produced from the manufacturers, like the old classic tube amps. Check out what Fender was using in their lab:
http://tone-lizard.com/fender-laboratory/ A 4.5 Mhz analog scope, which is pretty dang lowly according to today's standards. But Fender designed, built, and serviced some of the greatest guitar amps in all of history, still not surpassed today with all the advanced technology available. I guess that is 'last century'.
Being a noob myself to all of this, I can only recommend a couple of things: Get yourself a set of cheap probes (~$20 for a 100 Mhz set) from ebay, or whichever probes are recommended around here, for trying out whichever analog scope you decide to get. If you buy from ebay, be patient throughout and be courteous to sellers, but don't be a pushover if you run into problems. Ask the seller questions about the functional condition before buying, letting the seller know that you are looking for a fully working scope. You may still run into problems (I definitely did), but in every single case the seller was willing to work things out, being reasonable to more than reasonable. Good luck.