Yes, it is normal to feel scared. It is a new experience and you don't know this person well yet! In the past you probably have felt scared in certain important relationships when alone; you remind me of the first time I went on a date with a guy and we went to a fancy restaurant; I felt like we shouldn't be there, our parents should be there with us, worried he might not have enough money to pay, worried I would order something too expensive and my stepmother would find out, etc. It is very hard doing new things for the first time. Congratulations on your relationship and the hard work you are doing.