The crux of the problem is that to remain 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 would 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 D&D 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: what is the best way to proceed?
What about at different levels of experience?
How do you know when its best to split up and stick to your character?
In the backstory I mentioned a bandit group my character used to belong to but has moved away from. When searching a different bandit we had encountered and defeated I uncovered a note about me. There was another NPC who sent us out to face those bandits. In character this makes me suspicious of that particular NPC and wants to confront him alone (so as not to give away to the rest of the party that he was once a bandit).
As a player, however, I'm aware there is a larger quest we're set up to investigate in town and the other character would be focusing on that.
The session ran smoothly, I followed the DM's advice and it turns out that (given the discussion I'd had with them) they had planned for the split and worked an interesting join in - it ended up showing how the problems we were facing tied together.
In this instance the split worked but the advice in the answer below works well for general guidelines. My only add on would be to discuss it with the DM first, it might be that it plays into something they've got planned - if it doesn't then don't split the group.