If you are thinking about changing tone then you must be well acquainted with the tone of your present set up and have identified changes that you want to make in you systems tone before you start rolling tubes. Once you know what changes you want to make you should be able to communicate your desires, either in a forum or to a tube merchant, and get some reasonable recommendations. Without this knowledge you are just going to be spending money and time following generic recommendations from folks who don't have a clue as to what might really work for you in your system.
Getting quality low noise tubes is easier - tubes can be tested for noise by the seller (most sellers anyway). Getting more power is easy - there are only a couple of tube types (KT120 & KT150) which offer this possibility and then only if your amp can use them. Most can't.
Personally, when tubing an amp I start with power tubes and then fine tune the tone by changing small tubes. But that may be a bit advanced for some and it is easier, and cheaper, to play around with small tubes when first starting out if you just want to see what it is all about.
Re your questions about how to hear differences between tubes - you are right about recycling amps on and off too quickly/often, and it does take new tubes time to break in. Small tubes sound best after 24 hours and big tubes can take up to 100 or more. As you can see, you really do have to develop a memory for your systems sound in the first place. And develop an objective.
Lastly, don't get taken in with some of the BS about NOS tubes being the finest, etc. A lot of them may be the finest, but because of that they are usually hard to find and very expensive. There are many new production tubes appearing on the market which are relatively inexpensive and very good, if not better than many NOS, if you know what you need in the first place.
Take your time, it's a jungle out there. :-)