Your specific question here is how you can rebuild trust. That is always a challenge but more so when you are not together and he doesn't have to be transparent or work at it himself like he would if you were reconciling.
My ex continued to lie and lie for about 6-8 months after we separated. I gave him opportunities to lie. We were thinking a bit about reconciling so I did ask questions about personal things--and he continued to lie.
One thing Mom's House, Dad's House recommends is establishing a business like relationship. There's a really fine balance here--you need to separate from him for the sake of your sanity but also maintain enough ties that you have influence on parenting issues. Think business. Stick to facts--don't ask him personal questions and don't share your own emotional life with him. He doesn't deserve it. This also removes a lot of opportunity to lie. I think boundaries are a really good thing--and you can figure out how to do it so it doesn't feel too harsh. If he hangs out with her at your house, I would leave for awhile. If you have a phone, I wouldn't share where or what you are doing. Or work on chores in another room--I would suggest against hanging out together in the house. It will make you really, really sad and make things fresh again. There is also a steep curve to grief--much of it for me revolved around the loss of the family unit.
It is a really confusing time. You'll sort it out. Do that work on yourself.
And trust...I wouldn't trust my ex still with many things.
I do trust him with the kids though. He doesn't always make choices or follow through the way I would like but I can deal with that--happens even when people are together. He certainly spends more time with them now than he did when we were together.
So I would say that trust isn't something that you have to have in some ways. It is something he has to earn so the ball is not in your court there. Forgiveness on the other hand is in your court. Forgiveness is something you do for yourself and it really takes the emotional charge out of things. It's a process though. Steps forward and steps back.
I'm starting to ramble. This is a topic that has created so much growth for me and I clearly remember those early days of acute pain, acute confusion, acute anger, etc. It does pass when you work through it all.
You may find yourself like me--drinking a cup of coffee one morning with a big smile on your face, thanking the universe for what happened. My life is so much richer than it was. I feel incredible.
Do I think he should have been honest? Yes, and I do think some character flaws there kept him from telling the truth. And it was incredibly selfish--and I'm okay with a lot of "selfish" acts--but his affair did not take my well being or the well being of the kids into account. Good people do sometimes do bad things. Bad people do more bad things.

Book suggestions:
*Not Just Friends by Shirley Glass (will help you process the affair)
* Forgiving the Unforgivable (really helps to have some more info on forgiveness. I didn't find it to be something that I had been taught how to do and it didn't come easily for me)
* Too Bad to Stay, Too Good to Leave ( will help you get over any ambivalence you may have about being with him and help you not romantisize (sp?) things)
* Journey from Abandonment to Healing by Susan Anderson (helps with some of those feeling of abandonment and grief)
** website
www.survivinginfidelity.com support for infidelity with some nice folks who have seen it all
And my mantra became "living well is the best revenge".