You have now made it clear that you are seriously interested in each other. However, on your second date, you should keep a low profile. If you both feel like it, you can hold hands. Dress up a little, and pick a more formal date like a visit to the museum, the cinema, or a sporting event. On a second date, you have clearly communicated that you want to get to know each other better.
Is the chemistry right? Then you can go on a second date. Dress nicely, but don't overdo it, and enjoy a casual chat with a nice person. The first date is about finding out more about your interest in each other. However, there are some traditional guidelines that you should at least keep in mind: Dating rules in the USA Rules for first, second, and third dates in the USA can vary depending on cultural background, individual preferences, and expectations.