My son is about the same age and was writing his name in the beginning of this school year. They are starting to learn the spelling of last names, verbally, and tracing letters and numbers, along with trying to write them. I see a pretty big range of fine motor skill development in his class, so don't worry too much. Your dd will let you know when it is time to learn those things. If you push it before she is ready, she may rebel and refuse to even try. This could set up a bad feeling toward learning. Be gentle and play. Start with the first letter of her name, mommy, daddy, sister. Read, read, read, read, read......together.
The preschool has certain things they bring out in order to teach what is age appropriate, let them lead the way if she is in preschool and just reinforce it at home. Have her spot things/letters in books that begin with her initial or family members initials (meaning mommy, not actual first name). My son is starting to learn letter sounds, he is interested and we play at home with the sounds. They are also learning them at school. Most of the kids in his class have turned 4, but he has a late summer bday. Some of the 4 yr olds are getting it, some not. Some of the not yet 4 yr olds are getting it, some not. It just depends on your dd. It is a range of time when these things are learned, so don't force it.
My son loves this placemat he has that has the alphabet on it. My SIL gave it to us when my dd was younger. She outgrew it and now he loves to say his letters as he points to them and is playing with the sounds, too. It is always there and he initiates it. That is the important thing, let her initiate, but have it available. As simple as a placemat. Have fun and play, play, play.
Oh, and there are some 6 yr olds in my dd's kinder class that still have trouble writing their name. Dexterity with fine motor skills is a funny thing.