Which questlines should I do without a follower?

Honestly you can do any, all or none with a companion. In my experience, there are no specific questlines that should be done without a follower, but there are obviously some story quests that have to be done without a follower. In those cases a message appears saying "Lydia has headed home", or in some of the Dragon Born quests when you enter an alternate dimension, she stays with your body, while your mind wanders the alternate dimension.

As there is no real flow in the story in Skyrim, (quests can be halted every time you want to go fight a dragon or hunt some bunnies) a follower can be picked up at any point in the game. I personally have played once with Lydia, obtaining her at the expected point if you follow the first quest through from start to finish; once with no follower at all; and once with (the only) Argonian follower which I headed out for as soon as I got out of Helgen.

The only immersion-breaking experiences I have had with followers is, well, really all of them. Their AI is not completely stupid, but it still leaves something to be desired. They get in the way, whip their weapons out in all the wrong places, (sometimes going on rampages when you're on an assassination quest) and get themselves downed just as you loose a spell or arrow in their general direction, effectively forcing you to reload a save just to keep them around. (Huzzah for the Companion's Insight perk!)

That being said, they are excellent packhorses.

So it's really just down to personal opinion. It's nice to have someone who has your back, and likes to chat every now and then, but if you can deal with all the stopping and starting and awkward shifting around in narrow hallways.