Is it normal that my heart rate increased quickly after soda?

Yeah, even a small amount can make your heart rate jump pretty quickly. Some people are also just more sensitive to caffeine, but it’s still normal.

How soon after? It would have been a small amount, but also absorbed over maybe 15-30 minutes, depending on what else your stomach was dealing with at the time. I think it's as likely that your system sensed a gas bubble from the fizziness and that may have triggered either a Vagal response or the fullness sensation may have triggered an adrenergic response...I'm not sure, and only guessing of course. Or, your heart raced for another reason unrelated to your intake of soda. You can always attempt to repeat the incident and see.

I think the reason why Is increasing after having something with caffeine is the amount of caffeine. Some of these drinks are very high and caffeine and I do feel there will be some medical issues down the road because of