My 2003 Nissan Sentra was doing fine a few days ago until I started up a very steep hill and went into 1st gear. There was a burning smell for about 2 blocks so I shifted to 2nd gear. The smell went away. When I was in 1st gear it did not sound or feel like I was trying to go too fast for the gear. I just smelled the burning. Same hill in 2nd, no smell. How much should I worry?
A slipping clutch can definitely make a burning smell, but I would expect this to happen in second gear before first gear.
A completely different possibility is that you have a valve cover gasket leak that dripped oil onto hot parts of the engine when you first angled the car steeply uphill. Did you see smoke at any point?
No smoke, just the smell.