It's my first time being a DM and i was wondering for the sake of story progression if the DM was allowed to decide the turns of the players. Like am I allowed to say who goes when outside of battle?
Yes, you as the DM always control the tempo.
Usually in combat, that's using initiative order. This is strict because character stats/abilities directly bear on their initiative which bears on their survival, so "breaking initiative" is often very poorly thought of in D&D (as opposed to more narrative games). You could do it, but you almost never should.
Outside combat - you can use initiative order in specific situations (see Are there any official rules for or even suggestion of using initiative order outside of combat? for times this might make sense), or "go around the table and take turns," or let the group just act as the players speak up, with possibly you prompting those not getting much spotlight time - "hey Fred, we haven't heard from you for a little while, what is Kurgoth you doing?".
Usually outside of combat, when the group's together, you'd simply let the group freeform it, without any clear "taking of turns." You would only step up the levels of structure (prompting -> taking turns -> initiative order) when events are getting more tense or fairness in ordering or spotlight time sharing seems important. It's within your control as the DM to do this at any time, but don't exasperate your players by doing it all the time - you don't make everyone take turns in a friendly discussion around a table, and overusing imposition of structure on your players will turn into a problem.
If the group is broken up, then you also perform a variant of this in terms of shifting the "camera" on the scene to character/sub-group 1, then character/sub-group 2, and so on to make sure everyone gets spotlight time to do what they want. See How to effectively divide spotlight time with a split party?
In our games, when we're loose in town and headed out the next day to adventure, normally we'd kinda take turns...
DM: "What's Zartan doing?"
Zartan's player: "He's going to the magic shop."
Kurgoth's player: "Oh, Kurgoth wants to come too!"
Silentstab's player: "Silentstab wants to go make some criminal contacts."
DM: "OK, we'll do the magic shop first. Zartan and Kurgoth walk into a magic shop..."
Once we're in a scene, we'd usually just freeform.
Shopkeeper: "Hello adventurers! What can I sell you?"
Zartan: "I need some wands..."
Shopkeeper: "Well here's my stock..."
Zartan: "Is this unicorn horn?"
Kurgoth: "HA HA UNICORNS! When you're done with him I need a sword for slaying unicorns!" Shopkeeper: "I have just the thing..."
But if things start getting tense or "rulesey," then you start in on more structure.
Silentstab: "As we arranged, I pick the lock on the back of the magic shop while the shopkeeper's dealing with those two!"
DM: "Sigh. OK, roll Pick Locks." (Yes, that's very 1e, whatever)
Kurgoth: "I start bellowing about how unicorns are a scourge upon honest folk, trying to make a lot of noise to cover Silentstab's breakin!"
Silentstab: "I make my Pick Locks. I sneak into the back of the shop and steal everything and leave and go home..."
DM: "Hold your horses there buddy, step by step. Let's take turns. Zartan, what are you doing?"
And then when it gets real,
DM: "A magical alarm goes off! Maybe check for traps next time. The shopkeeper stands bolt upright and whips out a wand! Everyone roll initiative."