there are two ways to upgrade from this version. One is to download upgrade files and SQL query and follow instructions, other is to use KIM or upgrade utility. I prefer using first option up to version 6 and then continue with KIM (Kentico installation manager).
Here is your upgrade path:
5.5 > 5.5R2
5.5R2 > 6
6 > 7
7 > 8
8 > 8.1
8.1 > 8.2
For every version until 8.1 (where you have utility for that) you will need to manually upgrade you custom code and transformations. Best way would be to do this for every version:
1. Backup files/database
2. Run upgrade (you may also need to upgrade .NET framework on version 5 or 6)
3. Update custom code and transformations
4. Verify that site is working before moving to next version
Follow instructions in upgrade packages and refer to documentation. This is time consuming process. If you have lot of custom code it will be lot of work.