Noilani,
You say you have moved on, however, please know that Grandma's house is your daughter's home for now.
Kids often do cry during such custodial transitions. How long have you been at your new place? Does your daughter have her own room or space there? That is something that's very important to children, because it sends a powerful message that they *belong* to where they are going. Having a room or special dresser/play area is a way of saying "I have made a place here for you...you are welcome."
Kids need a lot of support. If you have a good relationship, do try to work with your daughter's grandmother to help make the transitions as smooth as possible. This means that, even if there is conflict between yourself and your daughter's dad or grandmother, those discussions are not taking place in the presence of your daughter.
If you do decide that you want custody of your daughter, then work toward that end. You don't say what sort of job you have, but if you have a regular schedule, working during the day and caring for your child in the evenings can be done. Single moms do this each and every day. Perhaps this would be a good time to consider your long term plans and goals, and what you want to work toward. Is there a child support agreement in place, and if so, who is paying who?
Above all, remember that your daughter's emotions are hers, and that--while they are painful or uncomfortable to you, they are not wrong. Four is very little to have to do this back-and-forth, so it's up to the adults to make those transitions as smoothly, as friendly, and as confidently as possible. It's up to us to contain our own emotions and allow the child to have their feelings, because they are the ones who truly had ZERO choice in the matter. Be sure NOT to tell your daughter that her crying makes you sad or upsets you. Instead, do try to give her some empathy. "I see you are so sad right now. You really like being with Daddy and Grandma, huh? (acknowledging her distress without judgment) We are going to come back in X days, and you will see them again. (reassurance.)" She needs to know her feelings are acceptable by you--this will help her to feel safe and supported. Remember, this isn't about YOU, this is about her signalling that the transition is upsetting. Once she gets permission from you to have feelings and some empathy, you can also reach out. "I know you are sad, and I am still glad to see you."
Be sure, when you are together, that she is your one and only priority. No boyfriends or friends over, hanging out. Play games with her, color and draw, read books together. Take her to the library and let her pick out a few books to keep at your place. Create small rituals which will become specific to her time with you. (is there a favorite simple meal like mac-n-cheese that she likes? Or light a small candle while eating dinner-- clink your glasses and do 'cheers!' before taking a sip) As a preschool teacher, I can tell you that children thrive on rituals and routines, so be sure to keep her schedule of meals, naps, etc. in synch with the one she has at Grandma's house. Rituals create a sort of psychological 'nest' for children; it creates an ethereal 'space' which allows us to rest, gives familiar form and shape to the day and all of this so supports the child.
Little by little, as you do these things and parent with compassion and understanding (please don't let her run roughshod, do be consistent with discipline), she will adjust to this. Much like daycare, some kids do cry at the drop-off/pick-up transitions for a while, but they do get over it after a while. I think part of this is our acceptance of their feelings-- that we CAN handle them, that those feelings, however unpleasant, are not too big for US as adults. (Because if they are too big for us, what chance does a kid have-- that sends a message that the child is unacceptable and they feel their feelings are too big even for them.)
Keep consistent, create a loving space in your new home for her, and look into your options going forward.