Yes, you can use clicker training while using other training methods. The purpose of the clicker is to mark good behavior. If you do not want to use a clicker you can use "good dog" but a click since it is a short sound gives the dog a more precise idea of what it is doing right. What method is your trainer using?
You can learn clicker training on your own. Name response is the first thing that I work with my dogs on using the clicker. My instructor does not have the dogs warm up to the clicker instead we just start by saying their name and when they start to look over click and offer the treat. Within 10 clicks or so they realize a click means a treat is coming.
In agility, I sometimes find I am not coordinated enough to deal with the clicker, the treat, and the footwork required so I may just use a "good dog" but they know a treat is coming.