However, should I get a contact me when I get in the tower airspace, or am I supposed to contact them manually when established? What about those times where there's no ILS/glideslope? Ended up disconnecting on approach because I was trying to figure out whether to just contact tower manually now or wait for a contact me.

Joshua,
Thanks for the reply, very informative.  To answer your question regarding contacting tower or waiting for a contact me...

Assuming approach and tower are online, typically approach will say something like, "Delta 123, Contact tower on 126.55" when you are 5-7 miles out.  Your reply to approach would be something like, "Contact tower on 126.55, have a great night.  Delta 123."  At that point tune your radio to 126.55 and contact tower with, "DFW Tower, Delta 123 ILS approach on runway 17R."  Tower will then reply with, "Delta 123, DFW Tower, wind 180 at 9, runway 17R cleared to land."

Now in your situation where approach was offline, or they went offline while you were in approach's airspace, you should switch to UNICOM on 122.8 and announce your intentions to other pilots in the airspace.  Skip this if you are already on final.  Once inside tower's airspace, 5-7 miles out (anything under 10 miles out is generally acceptable), contact tower with the same phraseology as mentioned above.  Usually tower will not send a Contact Me until you are too close for comfort and have not established communications with them, typically when less than 3-5 miles out.  In the real world there's no "contact me", so for your safety and others contacting tower is your responsibility.  Contact Me is just a VATSIM thing for controllers to get the attention of simmers that might be on the wrong frequency or not paying attention.

If there is no ILS approach on the runway you've selected, if you are not established on the ILS, or if you are flying VFR, contact tower with something like, "DFW Tower, Delta 123 visual approach runway 17R" or "DFW Tower, Delta 123, 5 mile final runway 17R".  The point here is you are letting tower know that which runway you're heading to and that you are on final within their airspace.  Tower will know you're there already, you're just reminding them that you're there and you are on frequency ready for further instructions.  Usually you'll get cleared for landing immediately, or something along the lines of "continue approach to 17R" if you are a ways out and tower is dealing with other traffic at the moment.

Congrats on your first VATSIM flight.  No need to disconnect when things go a little off script.  As a controller myself we deal with hic-ups constantly and are trained to control traffic in the safest and most efficient manner.  If things are truly out of whack and present an immediate danger the controller will tell you to go around and fly the missed approach procedure (either published missed approach or give you further instructions).  It happens often on VATSIM due to the nature of most sim pilots who are there to learn and further their knowledge of ATC procedures, so don't feel bad or take offense if you're told to go around.  In some situations it may not even be your fault that you're instructed to go around.  Again, adding a short note about your visual situation to your flight plan remarks section will alert all controllers along your route, and they will be very understanding of any situations that may arise and how to properly handle you and other aircraft around you.  Don't be sacred of ATC, we're here to help you learn and provide as real of an experience as possible.  Controllers on the VATSIM network encounter inexperienced pilots all the time.  I can all but guarantee with your vest for learning ahead of time that with our without any visual impairments that you are already ahead of the game compared to some of the other pilots we encounter on the network.

Continue your education in ATC phraseology and never be afraid to connect to the network and practice what you've learned.  That's exactly what VATSIM is here for.  VATSTAR, among many other VATSIM pilot training organizations, has some great resources for learning ATC phraseology, and you can even certify you at various levels of pilot competency.  I highly recommend reviewing the information contained their lessons and even certifying on the various courses if you are interested.

I've thoroughly enjoined your comments here in this thread, and look forward to controlling you in the skies if you're ever in the ZFW airspace.

Joshua,
Welcome to VATSIM, we're glad you and your friends are here and enjoying all that VATSIM has to offer.  I have a question for you, and it's in no way intended to be offensive, just curiosity for my ignorant self.  What enjoyment does a blind pilot in a computer simulator get from flying?  I question this since in a sim you don't get the feeling of motion you would in a real aircraft, and begin blind you don't get the visual effects from the sim application itself.  You're not the first blind person I've read comments from, there are even forum postings on the VATSIM ATC forums from a blind person interested in ATC controlling on VATSIM.  Again, my curiosity has the best of me in that situation as well.

While VATSIM rules do prohibit spawning up on runway, or slewing from ramp to runway due to potential traffic hazards, I'm sure that adding a short description of your situation added to your flight plan remarks section will alert the controllers handling you, and they will surely accommodate whatever you need to do to have an enjoyable flight.  If you need to slew at a controlled airport, I'd suggest talking to the controller and asking if it's safe to slew before doing so, just to avoid "magically appearing" at a runway hold point in front of another taxing aircraft. 

I'm glad you are here and enjoying flying.  What an inspiration to those of us that can see and get frustrated when a sim's graphics occasionally glitch.

Cool, thanks for the suggestion Brin.  Duet is working perfect for me, but I might try Spadedesk just for the fun of it.  Thanks everyone for the suggestions, it's save me a ton of development time.   

If it would require you to make a separate connection to VATSIM in order to use it, it would be against the Code of Conduct to use it at the same time as your ATC or pilot client.

It would.  I wasn't aware of that rule.  Oh well, it was a fun idea while it lasted.  Thanks Robert.

Holy crap!  So bummed I missed this.