regelmäßig
regularly
Happening or recurring at consistent intervals of time.
Er geht regelmäßig zum Sport.
He goes to the gym regularly.

regular
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regularly
Happening or recurring at consistent intervals of time.
Er geht regelmäßig zum Sport.
He goes to the gym regularly.

The word is made up of 'Regel' (rule) and the ending '-mäßig' (-like / according to). So it literally means 'according to a rule' or 'in accordance with a rule'. This origin explains both meanings: something that happens according to a fixed time rule, and something that follows a rule of form.
Die Kacheln an der Wand bilden ein regelmäßiges Muster.
The tiles on the wall form a regular pattern.
'Regelmäßig' (regular / regularly) can stand before a noun as an adjective or describe a verb as an adverb. As an adverb, the form remains unchanged.
Ein regelmäßiges Training ist wichtig für die Gesundheit.
Regular exercise is important for health.
In the time-related meaning, 'regelmäßig' (regularly) is often combined with activities. Typical phrases are 'regelmäßig Sport treiben' (to exercise regularly), 'sich regelmäßig treffen' (to meet regularly), or 'etwas regelmäßig überprüfen' (to check something regularly).
Wir treffen uns regelmäßig zum Kaffeetrinken.
We meet regularly for coffee.
When it is about shapes, patterns, or structures, 'regelmäßig' (regular) means even or symmetrical. You often find it in descriptions of geometry, art, or even biological processes.
Das Spinnennetz hatte eine fast perfekte, regelmäßige Struktur.
The spiderweb had an almost perfect, regular structure.
day
'Tag für Tag' and 'regelmäßig' both describe something happening repeatedly. 'Tag für Tag' means more specifically: every day, without much of a break. 'regelmäßig' is more general. It can mean every day, but also every week, every month, or at other fixed intervals. If you really mean every single day, 'Tag für Tag' is more precise. If the only important thing is that something happens again and again according to a schedule, 'regelmäßig' is more natural.

Ben geht regelmäßig zum Training, meistens dreimal pro Woche.
Ben goes to training regularly, usually three times a week.
Tag für Tag übt Mehmet zehn neue Wörter.
Day after day, Mehmet practices ten new words.