How omnipotent are they? Do they constantly wander the plane in an avatar form or are they sometimes ethereal, floating above the plane as an observer? Do they concern themselves in the everyday lives of their patrons or do they mostly only involve themselves in significant events? Do they observe everything or if they're focused on something over here do they miss that other thing over there?

Just trying to get a feel for DnD religion as I'm only used to the traditional roles of IRL religions.

First pick a world.

What do you mean? Is it different from Phandelver to Ravnica? What about for the Forgotten Realms? 

I'm just trying to get a sense for how they interact with the world.

Yes, it's a major factor for the campaign setting and can vary wildly.

In Theros, they are active and can be seen stomping around on the horizon.

In Eberron, they are distant and much less relevant in day to day affairs.

In Dark Sun, they are all dead.

Other campaign worlds tend to fall somewhere in between those.

And even within those generalized world-setting levels of activity for gods, it's up to the DM if they are that active, or more, or less.

Quote from scatterbraind >>

Yes, it's a major factor for the campaign setting and can vary wildly.

In Theros, they are active and can be seen stomping around on the horizon.

In Eberron, they are distant and much less relevant in day to day affairs.

In Dark Sun, they are all dead.

Other campaign worlds tend to fall somewhere in between those.

 In Eberron, they may or may not exist.

In the Forgotten Realms, they definitely exist and take an active interest in things that happen in the world that pertain to them, but their ability to directly act is limited.

I think it comes down to as mentioned the published setting if you're using one, but mostly what you would want as the DM.  In the IRL mythological pantheons (as opposed to the major religions practiced today ... though there are some parallels) D&D gods are largely derived from, the gods were in D&D terms, mechanically inconstant.  Their powers varied from god to god and fluctuated individually from story to story.  It's up to you and the game you and your players want to play as to whether the gods are background forces that give the world its shape, or "walk the world" either covertly or overtly, whether gods are an unstoppable force, or can they be killed (and if so, what happens when one dies?).  

Like in the world Exandria as stated in the setting book the gods locked themselves away from the world to put an end to its devestation by the gods war.

Quote from JayCharisma >>

How omnipotent are they? Do they constantly wander the plane in an avatar form or are they sometimes ethereal, floating above the plane as an observer? Do they concern themselves in the everyday lives of their patrons or do they mostly only involve themselves in significant events? Do they observe everything or if they're focused on something over here do they miss that other thing over there?

Just trying to get a feel for DnD religion as I'm only used to the traditional roles of IRL religions.

 Most gods are not very omnipotent and only rarely manifest in avatar form, but like @scatterbraind said it varies from world to world. Forgotten Realms and Dragonlance both have a single bigender or agender God and a bunch of lesser gods of various alignments.

It also depends on the time the game is set in. I've played in a no/low magic game set in the Forgotten Realms, during the "Time of Troubles", when all the deities became mortal and divine magic ceased to exist. 

Quote from Sigvard_Vigrid >>

It also depends on the time the game is set in. I've played in a no/low magic game set in the Forgotten Realms, during the "Time of Troubles", when all the deities became mortal and divine magic ceased to exist. 

 That’s true. I read those books.