Learn Romanian: Fac gimnastică în fiecare dimineață.

In Romanian, time expressions with fiecare (every) usually need the preposition în (in) to show when a recurring action happens. So you literally say 'in every morning'. You will also see this in phrases like în fiecare zi (every day).

Does gimnastică mean I am doing flips and cartwheels, or just regular exercise?

In everyday Romanian, fac gimnastică usually just means doing a light morning workout, like stretching or bodyweight exercises. If you want to talk about going to a fitness center or lifting weights, you would say merg la sală (I go to the gym).

Does the word fiecare change depending on whether the noun is masculine or feminine?

No! Fiecare is an invariable adjective, which means its form never changes. You use the exact same word for a feminine noun like dimineață and a masculine noun like an (year), as in în fiecare an (every year).

Why doesn't gimnastică have an article like 'the' or 'a' attached to it?

When you talk about doing a general activity or sport with the verb a face (to do), you use the plain, indefinite form of the noun. Just like in English, you say 'I do gymnastics', rather than 'I do the gymnastics'.