Formal:
* Appoint
* Nominate
* Elect
* Assign
* Name
* Specify
* Indicate
* Denote
* Establish
* Define
Informal:
* Pick
* Choose
* Select
* Set
* Label
The best synonym for "designate" will depend on the specific situation.
For example, if you are talking about choosing someone for a job, "appoint" or "nominate" might be better choices. If you are talking about setting something aside for a specific purpose, "assign" or "designate" would be more appropriate.