ID in alt text. Direct link to the original artwork in the source of this post.
One of the things I appreciate most as a dungeon master is when my players not only RP (I love you for it), but immerse themselves to the degree where they try to avoid mechanical terms like "short rest" or "skill check". So here are some ways I've found that can be done, if you really want to treat your game master/storyteller/person who runs the game to some immersion they don't have to be responsible for!
Instead of saying "I want to roll [skill] to..." (because, you don't get to choose your skill check - the game master does), instead explain what it is that you want to achieve:
"Do I believe what she's saying about her family?"
"I want to look for signs of hidden doors in this room."
"I am going to take the device apart to see if I can figure out how it was made."
"Does the painting look like anything I can recognize?"
Doing it like this means you hand your game master a springboard on a silver platter (unsure if that combination of metaphors work, but whatever) from which they can easily jump into a description of your efforts.
If you work with a system like D&D where you are encouraged to take frequent in-game rests, it's a lot more immersive to talk to your fellow players about your needs for rest if you do it in character, instead of saying "I'm out of spell slots" or "I want to take a short rest". Here are some suggestions:
"I really need a moment to sit down."
"Don't know about the rest of you, but I have some wounds I have to tend to."
"I can't go on like this, we need to find somewhere to sleep for the night soon."
"My head hurts from casting spells all day, I think we should try to avoid fighting until we've rested. Is there an inn nearby?"
After you've discussed it in character, either wait for your game master to ask whether you're taking a break, or tell them outright. You should always confirm that you're on the same page, but just a bit of in-character buildup is something I personally really appreciate. It also adds to the feeling that your characters are real people experiencing exhaustion, not just (video) game people.
You can't see hit points! They're an arbitration for us as players to keep track of how well a character is doing physically, but asking in the middle of combat "who's low on hit points?" when that is not something the characters know what is kinda wrecks it for me. Instead, consider:
"I saw [character] was struck earlier, how badly hurt do they look?"
"You see how blood is pouring from a deep cut in my arm, it looks really painful."
"[Character] is a little woozy from the blow to the head he just suffered, but he'll be fine."
"I want to spend some time examining the burns [other character] suffered during our last fight - how bad is it?"
Basically, treat your player characters' hitpoints the same you do the hitpoints of an enemy: With zero idea of what the actual number is.
Not gonna lie, unless you are into martial arts and such, you probably run out of ways to describe "I hit the monster with my sword/fist" fast. However, just going "I roll to hit; that's [x] points of damage" again gives that very video game-y feel that I think should really be avoided as much as possible when you sit with a TTRPG. Try one of these on occasion to shake things up:
"I take a running leap and bring my axe down hard on its arm!"
"With my legs around her throat, I slam my elbow into her face!"
"I do a series of somersaults toward the mafia don and try to kick him right in the gut!"
Just because you can't cast spells doesn't mean you can't pull some theatrical moves!
A lot of what I said about melee combat above also applies to ranged combat, I feel - and can be solved similarly.
"I point my gun and aim for the tiny gap in their visor."
"Nocking one of my enchanted arrows, I shoot at their feet and try to ignite the ground around them."
"I aim for the rope holding the chandelier in an attempt to shoot it down."
"Bracing for the recoil, I aim my rifle at the hand she has her sword in, and pull the trigger."
Maybe look up some videos of people using the weapon(s) your character uses to get a feel for how to describe the movements involved? It's certainly something I need to do more!
Spellcasting is a special beast, not only good for combat but also in many utility situations. And there are a lot of ways to talk about it without referring to spell slots or class spell lists. Here are some:
"Can you cast [spell] today?" "No, I'd have to meditate on it/consult some books first."
"I bring out my magical orb, and you see how it starts to spin rapidly over my hand until it's just a blur, and I cast [spell]."
"I'm sorry, that sort of thing is beyond my capabilities. We should try to find a powerful wizard, they might know how to do that."
"Five lightning bolts shoot from each of my fingertips, right at the enemy!"
Take advantage of the spell descriptions that come with the spell to explain what it looks like when you use the various components, and when the spell takes effect - and maybe also what happens when it fizzles out for whatever reason.
If you're not super comfortable with roleplaying, but want to try doing it more, I hope you take this as positive encouragement and as an example that it doesn't have to be all that hard or elaborate all the time to increase your feelings of being immersed in the game. We can do this!
Disclaimer: 95% of my experience with TTRPGs are in fantasy settings. I'm sure someone else has much better examples of roleplaying for Sci-Fi and other genres. Please do add to this post if you have any thoughts!