I've heard drinking lemon water first thing in the morning helps to lose weight, but is it a good idea to have breakfast immediately after that?

Sure: Forget the lemon water. It you do not restrict carbohydrate and caloric intake, no amount of lemon water will help.

It's the Water...: It's probably the water, not the lemon, taken before any meal or snack, which may help you take in less calories by giving you a fuller feeling with less food. Tracking your calories with an iPhone app such as "Noom Coach" is very effective way to balance both basal and active calorie requirements with your dietary intake -- plus it's kind of fun for the somewhat obsessive.