From what I can see you need the appropriate pronoun, the appropriate version of zijn, + aan + het, and the (infinitive form of verb) to express the equivalent of I am __ing. If you want to say I am swimming, eating, drinking, etc. in English, you use the appropriate pronoun, the appropriate version of the to be verb (am, are, is), and the appropriate form of the verb (which always or almost always ends in -ing). Dutch does essentially the same thing. The main difference that you have to remember is that you use "aan het + infinitive" instead of the English verb ending in -ing (swimming, drinking, eating, etc.) Once you remember to do that it's easy! Although as yet, I haven't the faintest idea why Dutch does it this way. (Not that I know why English does a lot of the things it does, either, but back then I didn't feel as much of a need to understand how and why as I do now).
But "I'm riding to the capital" would be ambiguous, you could be riding on a horse too. The use of 'bike" as a verb is informal but correct English, and it may be more common in some parts of the English speaking world than in others. Unfortunately, there's no one verb in English which is as common as "fietsen" is in Dutch.