Welcome to the forum and congrats on whichever phone you decide to purchase.
With phones, everything is subjective. Of the phones you listed there really isn't anything that you won't be able to do with them. The one snag you may hit is the locked bootloader on the Razr. This will prevent you from installing custom ROMs. The HTC won't give you any problems in this department and I assume the iPhone won't either (not an iPhone girl obviously

).
When I am going to purchase a new phone the first things I look at are the UI, size and feel as well as how it does on basic tasks. If I can't stand the look or how the UI behaves then it is usually a no go for me. Even though I typically root all my phones I still like to enjoy the factory look and feel sometimes. If it is just ugly or hard to maneuver then I will stay away. The same goes with the overall feel and size of the phone. A phone that is too big for my hands (little girl hands) just isn't practical for me. If the buttons are in an odd place then it has the same effect because these are things that will never change. Last, basic tasks like opening the market or messages should be very quick and smooth. If the phone lags when doing these things then it isn't going to perform well with anything else either. This is something that should be tested after the phone has been restarted, BTW. You don't know what the last 20 people to touch that phone have done with it.
As far as what you plan to do with the phone:
Texting, make sure you like the keyboard and you are comfortable typing on it. There are plenty of 3rd party keyboards, but it is a good idea to at least like the stock one in the begining.
Browser, I would download speedtest from the market and see about the data speeds. I'd also open each phone up to the same sight and see how well it loads. Is the scrolling smooth? Does it zoom in and out smoothly? Does the browser offer a desktop mode? You may not think that last one is all that important, but it will come in handy quite often. There are always 3rd party browsers for Android and IOs, but again the stock browser should be able to perform well for you too.
Tumblr, Twitter, etc. These types of apps will run on most phones so it isn't really a performance issue on the 3 you listed.
Music.....This is were things get a bit different. The iPhone has iTunes and all your music will be going through that. If you like the software then it shouldn't be an issue, if you don't then......The Rezound does offer Beats, but that is only good if you like Beats. Some love it and some hate it or simply find it ok. I honestly wasn't all that impressed with it. The key here is that beats performs with the stock player on the Rezound and I'm not sure it works with other players. So if you like the stock player great, but if you don't then it could kill beats for you. The Razr I believe uses a different player and I don't have too much info on it. With the Android phones there are always 3rd party players or Google Music in the market. I personally love Google Music since it allows me to have my entire music collection with me all the time, but I have an unlimited data plan and streaming a lot of music can be a problem area. On any of the phones you will have the option for Pandora, Slacker, Spotify, Rhapsody, etc. to get your music fix as well.
With games it also a toss up between the three phones. There are plenty of games on each platform and each of the phones should work well with just about all of them.
Like I said in the beginning you've picked 3 really great phones. While I wouldn't rule out the Galaxy Nexus (I have it and love it), that is completely up to you. I understand this being your first Verizon Smartphone and not wanting to run into problems. Good Luck.
EDIT: I just figured I'd add one other thing. When you go to look at or purchase a phone don't listen to what others tell you in the store. Make sure they let you test the phone out yourself. Also, don't make a decision based on what any of the sales reps are pushing, even if that very attractive girl is flirting with you.

I am not saying that all reps are like this, but I have watched plenty of them lie to customers about a phones capabilities (plenty have tried it with me too). They are there to sell phones and not be encyclopedias on all the phones there. If they are telling you something that you have not seen or heard before, verify it first.