It all depends on the writers and how tired I am, lol.
Sometimes I'm really in the mood to get grabbed by a story and will invest the effort to read over things and avoid skipping.
Other times, I just want to play through and will kinda click through the text (especially with the fetch quests in FFXIV). But I know there is good lore in there if I look for it, so I feel bad sometimes.
I never skip cutscenes, though. I try to follow those as well as I can.
The ones I do love but have trouble playing are visual novels, though. I often want to get into the story but the lack of gameplay can sometimes make it feel tedious. Depends on the game, of course, but I really have to make a lot of effort with VNs.