Story in an RPG comes out of the choices the players make in response to the world. DnD enables that just as much as any other system. And nothing about putting yourself in your character’s shoes and making choices requires taking an improv class or talking in funny voices, so don’t worry about that.
But a bad DM who treats every non-combat encounter like a video game cutscene can be a bad DM in any system.
That said, like another poster mentioned, 4th edition did tilt heavily towards being a skirmish wargame dressed as an rpg. 5e is much better in that regard.
But if you want to get away from swords and sorcery fantasy and if Lovecraftian horror does anything for you, check out Call of Cthulhu. The rules are straightforward, creating characters is easy, and it usually focuses on unraveling a mystery rather than completing a Conan the Barbarian style quest. Combat is de-emphasized, because characters that keep getting into fights with unspeakable eldritch horrors tend to have short life expectancies.
If you haven’t given 5e a try yet, do so - it’s substantially less crunchy that previous releases and frees combat from the tyranny of the grid if the group wishes to play it without.
As far as balance between combat and story, that’s all up to content. Published adventures tend to be very heavy on dungeon crawls, while many homebrew adventures lean more towards RP.
That said, from reading about it, I do think that the mentioned Cypher System will be a better fit for me
https://startplaying.games/join-game/intro-to-monster-of-the...
https://startplaying.games/join-game/thieves-in-the-night-16...
Or Cypher System https://startplaying.games/join-game/black-rain-ripples-[cyp...
Both are pretty fun systems!
The Cypher System is probably closer to D&D, but the combat / rules / stats are a lot more streamlined, so in my experience it's easier to jump between narrative and action.
Monster of the Week is a bigger (and welcome!) departure from the traditional TTRPG formula, and is probably closer to what GP was looking for -- it's a very light "game" system on top of the players building a story together with the DM.
http://apocalypse-world.com/pbta/
https://www.drivethrurpg.com/browse.php?filters=44825_0_0_0_...
But in a pick-up game with strangers, and with the current "mainstream" d&d - you probably have to be lucky to enter a campaignwwhere you develop a character as rich as Elminister or the Lord of Waterdeep.
For systems that encourage story-telling - I'd look to Cypher as mentioned by others - and of course white wolf's Storyteller system. It can be used for almost anything - but Hunter and Mage are probably most beginner friendly for players. Beware that mage can be quite mind bending.
A wiki from our current mage campaign: https://radio-noir.fandom.com/wiki/Radio_Noir_Wiki
Old blog from a previous storyteller/mage inspired original setting:
http://thebookofworlds.blogspot.com/p/world-of-argos.html
Ed: I've also heard good things about:
https://en.m.wikipedia.org/wiki/Castle_Falkenstein_(role-pla...
And for a fun ruleset (but slightly dated setting) check out Underground! https://en.m.wikipedia.org/wiki/Underground_(role-playing_ga...
You're probably looking for something like the old Call of Cthulhu by Chaosium:
https://en.wikipedia.org/wiki/Call_of_Cthulhu_(role-playing_...
I see from the wikipedia article that the last edition was in 2014, so it's still going. There used to be a d20 version, also, for those people who don't know how to read two d10 side-by-side ;)
You might also want to check out the user forums at rpg.net:
https://forum.rpg.net/index.php
You can find threads for all sorts of games, more or less popular. You're bound to find something that you will like.
Also systems come with setting and genre - if you like medieval fantasy and political intrigue Call of Cthulhu might not be for you (because it takes place in more modern times in real-world+ universe and it's mostly focused on horror/mystery). World of Darkness has a culture of focusing on narrative and acting over mechanics in this system, but it can be a little too "emo"/"existential horror" for some people. Or maybe it's my prejudice, I've never personally played it, just watched some games in the 90s.
If you don't like crunchy mechanics there's a very simple d6 system:
1 = you failed AND something bad happened
2 = you failed
3 = you failed BUT something good happened
4 = you succeeded BUT something bad happened
5 = you succeeded
6 = you succeeded AND something good happened
Your character sheet is just a table with free-form description of what you're good at (you roll 2d6 and choose better result) and what you're bad at (you roll 2d6 and choose worse result). DM can also give you more/less dices depending on how hard something is to do.
That's pretty much it - you can play it now :). I don't remember what was the name of that system but it's commonly played late-night at conventions - almost no setup required. And it tends to create games with little focus on combat.
Voice-acting is a matter of personal preference I think. I don't usually do it and I've never had problems in games where some people did it and some didn't.
Anyway - the key to having fun is to set expectations on the first session. Better to talk about it in advance than to waste time playing with people who want to have fun differently than you. The system is secondary (and you can always houserule stuff away if it's too complex for you).
I have a question about which system is better for a story-driven game for beginners.
Some context: I like RPG, but I have played exactly 4 times in the last 20 years. Last time I played it was D&D and my experience was it was more of a tactical game focused on combats than characters and story. It was about the squared map, knowing all the rules of your weapons and spells and trying to beat the GM in this game. All the rest seemed like cutscenes to the combat mini-games.
I didn’t enjoy this that much, especially since I do not know all the rules, limitations and possibilities of my weapons/spells.
I would prefer to develop a story together. A system where you don’t need a exact map with squares/hexagons to make decisions, for example. Where story and character decisions on what to do matter more than tactical decisions in combats.
Also, despite living story-driven games, I am not much into “acting” (doing voices, emotions and reactions like my character). Not sure if I have too many restrictions to find a good system or group here.
So, which system should I look for that it’s welcoming to beginners and story-driven?