There are three expressions in English that incorporate the word first:
first at first firstly
These expressions are often confused. Each of them has a particular use.
We use first when we are talking about a sequence of events.
First, we opened the door. Then, we let the dog out. Finally, we brought the furniture outside.
We use at first to mean something like in the beginning or originally.
At first I didn't like her, but as time went on, I realised she was actually very nice.
Firstly is generally used to indicate the first, most important point in an argument or discussion.
Firstly, I think it's a bad idea. Secondly, I don't think it will work. Thirdly, we can't afford it.
Firstly can be replaced by first and foremost or the popular First of all.
Now, as long as we're on the subject ... When we are talking about a sequence of events, we usually use the word then (where Polish would use potem), NOT next or after.
First comes love, then comes marriage, then comes the baby in the baby carriage.
Next is more commonly used in instructions, such as recipes or directions how to use a device or piece of equipment:
First, mix two eggs. Next, heat some water in a pot. Next, ...