How can I get his trust back
I'm sorry to hear you are going through this. It seems like the only thing you can do when you break someone's trust is to come clean and then attempt to repair the damage that has occurred. You can seek an opportunity to rebuild the relationship, but unfortunately, if he doesn't want to, that is up to him. You can only control your own behavior, so you could use this opportunity to be a more honest person. Either you will be given an opportunity to fix this, or not. If he is open to it, coming to therapy might be helpful to open up a dialogue, but if he does not want to, that is understandable.
You might want to consider your own well-being in that you are in a one down position, and in a way, until this is cleared up he can always hold this against you. At a certain point, if you are not being given a chance to be forgiven and he is still spending time with you, you might want to consider if that is healthy for you, as it seems like it would be challenging to have a cloud like this hanging over your head.
The good thing is that you have admitted to your lies and come clean. You have the opportunity to never allow this type of thing to happen again, by being open and honest. If you are not given a chance to fix this relationship, that is unfortunate, but you also don't deserve to suffer indefinitely.
It seems like if he is willing to go on trips with you, then perhaps there is a chance that you could be forgiven, but if it keeps going and you are never given a chance to fix things, you may consider how long you want to be in limbo. That is, at a certain point you might want to move on if you are not allowed to fix things, or if he feels like he can never trust you again.
All you can do is ask for an opportunity to make amends, everyone makes mistakes and deserves a second chance. Unfortunately, since you have been the person at fault, your second chance might be with another person. Whatever you did, you can be forgiven, and at a certain point, you might be able to forgive yourself, as well.