Playing DnD isn't as difficult as the 300 page Player's Handbook makes it seem. As long as you know the basics, you can pick up the dice and start playing with a new group today! Below are a few sections going over those basics.
Everybody plays the game differently, and that's ok. You don't need to bring out the funny voices and dress up with mood lighting and props. You can be a wonderful player, and get a lot out of the game, by simply showing up (prefereably with snacks to share) and participating.
Some players prefer to simply state in their own words what their character will do. Others will act out the character's actions, standing up from the table and giving the character a voice of their own. Others will spend most of the game passively watching the story unfold arouns them, and others will be more involved in leading that story, actively engaging with every NPC.
Every play style mentioned here is completely valid. The only thing in Role Play that matters is taht you as a player are participating in the game. Tell the DM what you'd like to do in any way you feel comfortable, but you won't get anythhing out of the game unless you're playing the game. If you're uncomfortable role-playing at first, that's fine. As you play more you'll get more comfortable with it and, who knows, maybe you'll one day be the person who brings props and mood lighting!
As a player, you'll be dictating the actions of a single character. This means you'll need to know a little about the character. You also need to decide on the character's stats and know what class you're playing. The Player's Handbook (PHB) can guide you through the math of it, but here's the basics:
The end result is a character sheet with everything you need to know about your character. When roll-playing you'll use your skills and proficiencies, or lack of skills, in order to know what your character is like in conversation or combat. If you're ever not sure what your character would do in a given situation, all you should need to do is look over your character sheet.
Combat occurs in turns. Players and the DM roll for "initiative" at the beginning of combat to determine their turn order, by rolling a d20 and adding that to their dexterity modifier. Each turn lasts 6 seconds in-game, and most battles are over within a minute of in-game time. This, naturally, can take up to several hours of real-world time, depending on how much is going on.
Combat can either take place on a physical grid on the table, with mini figures representing characters, or it can take place in the Theatre of the Mind, where all players agree on a loose sort of combat that takes place purely in their imaginations, collectively. This works surprisingly well.
On your turn, you have several options of things you can do:
You typically have one Action per turn. This is usually an attack, or it could be casting a spell. It could also be attempting to parlay with your enemey, attempting to deduce something about the surroundings. An Action is anything that takes most of 6 seconds to do, and is the main thing you do on your turn.
Your character may have a Bonus Action to use. Not all characters or classes have Bonus Actions, so look through your character's abilities for one. If you have a Bonus Action, you can do whatever it is in addition to your Action on your turn. For example, your 5th level cleric can cast Sacred Flame as her action, then cast Healing Word as her Bonus Action, because Sacred Flame costs an Action to use, and Healing Word costs a Bonus Action. Bonus Actions are incredibly useful in allowing your character to do more stuff in battle, so check to see if you have any available to you.
Reactions are done at any time in a round, although you only have one per round. Like a Bonus Action, Reactions are specific things that your character can do in response to something happening. A Reaction will always have a specific trigger that needs to happen for you to take the Reaction. A typical example is when an enemy leaves your melee attack range. In this case, you can use your Reaction to get what is called an Opportunity Attack, basically a free attack while the enemy runs away from you. Reactions happen quickly and, depending on your DM, if you miss a trigger and gameplay continues without you taking your Reaction, you may have lost the opportunity to take it that round.
There are a few things that you can do that don't cost the above mentioned mechanics. On your turn, you can speak a short phrase to someone without costing anything. You can also interact with an object without using your action, as long as the interaction is small in nature. Something like opening a door or drawing a blade are examples of objet interactions. Object Interations allow the combat to feel more alive, so always look for things to interact with!
Your character has a movement speed, typically 30 feet. This means that on your turn, you can move up to that distance. If you're playing on a grid, every 5 feet of movement is one square on the grid. Things can get in your way such as obstacles or enemies, and moving through something like a pile of rubble will take double you movement. You can also break up your actions and your movement, so you can attack, move, use a bonus action, then move again. In this way, combat can be very dynamic, use this to your advantage.
Your skills and proficiencies dictate what your character can and will do. While talking with NPCs or crawling through a dungeon, always be looking at what your character is good at. This will let you be more successful in role-playing, and will help you to benefit the party. While crawling through a dungeon, for example, if you're a rogue you should be looking for locks to pick, or stealthing and scouting ahead. If you're a druid, you can best be looking for traps or asking animals if they're seen anything. The game is designed with a balance in mind that assumes players use their skills, so don't be afraid to use them. Of course while you should make sure to lean into your character's skills, don't be afraid to do something that you're not good at, feel free to make a medicine check with that barbarian if a great RP moment shows up!