I see two issues here. 1. the phase"at first" means "initially" or "at the beginning". It isn't followed by other actions that are numbered. You can say, "at first, he was small. Then he became ..." but not, "at first, he was small. Second, he became ..." 2. When numbering actions or points, there is some disagreement about "first/firstly", "second/secondly", etc. In my experience with US English, we tend to use "first", "second" etc. but you will heard the other forms as well. You have a list of actions, so: "First, I did my homework. Second/next/then (choose one), I washed the dishes. Third/finally/last/then I went for a walk."

"At first" is an adverb phrase that implies that something was conditional. At first, it was cold but then it got warmer. At first, I started my homework but then decided to listen to music. The phrase could be substituted with, for example, "To begin with." I sense from your sentence that you mean... "First, I did my homework. Secondly, I washed my dishes and then I went for a walk. ["at second" is not a correct term at all. If you are referring to a number of steps that you did, use "secondly"]. "The first" on its own is not an introductory phrase. In your sentence it could be written as... "The first task that I did was my homework. Secondly, I washed my dishes and then I went for a walk."

In describing a sequence, it should be either "First" or "Firstly' (no "at" or "the") and "second" or "secondly". The choice of which option to use is a matter of style; the "...ly" forms are a bit more formal and start to sound awkward as you count up to around five. "At first" applies more to changing situations than to a series of discrete steps, e.g. "At first, I was confused, but as he explained, it became clearer." Consequently, "at second" makes no sense.

At first