Balancing story with team play

The crux of the problem is that to stick true to my character and the situation he is in I feel he wouldn’t bring the rest of the party along to something he needs to do. Me the player, however, is aware that this breaks up the flow of play and other players will have to sit around for a while waiting.

I’ve spoken to my DM and without giving anything away they said I should feel free to stick to my character. We’re all fairly new to DnD though so I don’t want to put too much strain on us at this point.

There are plenty of people here with more experience than us, how would you proceed? Split up? Keep together? What about at different levels of experience?

How do you know when its best to split up and stick to your character?