Going to post about my experiences as a web GM for several years now and try to answer your questions. 90% of my players have probably never heard my voice.
If you want to learn D&D or any other tabletop game, it's very easy to get started! If you are new to RPGs, I would recommend D&D 5th edition, which also offers a version of the rules and handbook for free. This should teach you anything you need to know to run a game. If you want to know more about D&D or any other game, consider buying their "Core Rulebook" or "Player's Handbook"; these are always in book or e-book format.
Basic Rules for Dungeons and Dragons
The rules will give you the mechanics to run combat, character building and handling situations, but a Role Playing Game like this doesn't quite play like the videogames. It requires 1 player to be the Game Master (GM/DM) that handles all the NPCs and story while the other players (preferably at least 3) become the Player Characters (PCs/Party members). It's completely free-form and your character can try to do anything you want, making it much more open/free/sandbox than any videogame that exists.
Remember that all RPGs do not run in real time, but in turns or some time abstraction. Even if you are slow at typing, the game 'pauses' so you can make your decision whenever it's relevant like it was a classic videogame RPG. The speed advantage of using verbal communication becomes moot in this context.
As far as playing online, there is a lot of software that can accomplish this. Most popular ones I've seen are Fantasy Grounds, MapTools and Roll20.net. Of those, my favorite for sure has been Roll20, which gives you all the resources to play an authentic game online without needing any player to download any software.
Though dialogue is extremely important in an RPG, it can be conveyed in text form perfectly. In a real life situation where you are together with your friends it would be extremely difficult to communicate without talking, but in an online environment such as Roll20, the norm is to do it by text. The major downside of playing by text is that it can become pretty slow since the player engagement is not as high as when playing Live.
You only really need 4 ways to communicate in these games:
- Talking as your Character or an NPC. The character is speaking in the game. It's the same as when a character in an old videogame talks in that you should expect to see the name and avatar of whoever is talking. This means the DM doesn't have to 'make voices' for the characters either (Which is good if you suck at it).
- Talking out of character. For all the stuff that is not role playing, like talking with GM, describing things to the group, making jokes or whatever else. It can still be very fun to chat with friends like this.
- Describing the actions of characters. RPGs are supposed to mostly be in your imagination. You can get tokens on the field to show everyone's relative positions, but actual action has to be described in words, It's easily done in text as if writing a book or screenplay.
- Visual Medium for Combat. Doing text-only combat with just text is really hard, since this often requires a lot of information exchanging betwen players and the GM. A visual medium like a grid on a map with tokens representing all the creatures can speed things up a lot. A picture is worth a thousand words, after all.
All 4 are very easy to accomplish in text form online as you can switch your chat name at will (to any character) and use emote formatting for showing what character is doing an action. They also provide ways to roll all of the dice you need very easily, store character sheets and handle maps for combat and scenery.
If you have an online group of friends, I'd recommend you organize them to play on one of these platforms. Only the DM (Usually most interested player who organizes things) should really need to learn the rules of the game before starting (The PCs can learn as they go). You should probably organize them using some other group chat tool such as Skype/Discord/Slack/Hangouts.
If you don't have a group of interested players, you can still learn the rules and try to find a group online in a forum or reddit, Roll20 also has a way to search for "open" games looking for players.
If you still have any questions, you are in the perfect site to have them answered.