Spells don't level up. But, you have a few different ways in which spells get more powerful:
1. You can "upcast" some spells. This means you cast a level 1 spell as a level 2 spell. That means it takes a level 2 spell slot. Usually if you can do this, it makes the spell more powerful in some way, but it depends on the spell. A good example of this is Magic Missile:
At Higher Levels. When you cast this spell using a spell slot of 2nd level or higher, the spell creates one more dart for each slot level above 1st.
2. Some spells do automatically get more powerful based on your level. Fire Bolt is a good example:
This spell's damage increases by 1d10 when you reach 5th level (2d10), 11th level (3d10), and 17th level (4d10).
Quotes taken from the D&D 5th Edition Player's Handbook.
I don't think you were actually talking about Skills (Acrobatics, Athletics, etc) but those increase in different ways:
1. Via the stat modifier that they are tied to (Strength for Athletics)
2. Via your proficiency bonus (at the start, skills you are proficient in get a +2 bonus. This increases to +3 at level 5, etc)
