[BETA POST to get the content out there, will build out if there's interest]
1. Narrow down the options
Pick up as many racquets as you can (e.g. in a store), move them around, go through your swings, and then select a substet that appeal to you. And/or ask all your tennis friends who you can get a hold of if you may borrow one of their frames.
Pay attention to grip size too - they can feel different between manufacturers, models or even model years. In terms of size, factor in the overgrip if you use one.
If you're planning to identify a set of demos online, either start from the racquet of specs you know you like, or eliminate what you surely won't like - huge head size, super heavy, super light, dense string pattern etc. Tennis Warehouse for example offers a Racquet Finder (which I wish was a bit easier to use), and also shows similar racquets on each product page (RF97 example).
2. Try your favorites
Pick your favorite frames to demo, and maybe pick up 2 extremes that you think you won't like, just to get a sense for the spectrum and question your initial impression.
Any self-respecting tennis store will offer a demo program. You can also order demos online, e.g. at Tennis Warehouse, Tennis Express, or if you're in Germany for example, there's Tennis-Point.
3a. Found a clear winner?
Go for it. And enjoy it for the time being - if you discover that you don't like something, you can always adjust later. At least you got on court in the meantime :) If you just like to tweak your racquet a bit, you can find an idea here. If you're looking for a more drastic change down the line, select your future set of demos based on your current pick and adjust the spec(s) you're no longer fond of.
3b. Not in love yet?
Note weight, balance, swing weight, head size, string pattern, and any other spec that stood out to you. Even better, measure what you can. From there, figure out what you liked and what you didn't.
Then pick the next set of demos by keeping what you liked, and changing what you didn't. If you want to keep track and rate your demos, there's a template here's that you can adjust and use.
Hope this gives you some ideas so far. If you're an advanced player, here are some other things to pay attention to:
* When buying multiple racquets, get them matched to what you like and to each other. Over the years, I wrote a few posts about my matching woes - here's an example.
* If you're sensitive towards the grip, remember that there are different butt caps (somewhat easy to change) and grip shapes (not easy to change).
* If you know exactly what you want, you can order a custom racquet, e.g. at Head or Vantage. Fun playing around with this the Head customizer in any case :)