I want to create a program that outputs 5 even numbers starting with 2 in order from the smallest number.

In addition to the numbering variable n, define the variable limit for how many outputs you want.
Then, increase the limit variable 1 and turn it in the while loop.
In other words, while turns the loop and uses if to determine even and odd numbers.

n=1
limit = 0
while limit <5:
  if n%2 == 0:
    print(n)
    limit = limit+1
  n = n+1

Run Results

2
4
6
8
10

You can use the while loop, but we know that even numbers exist for every two integers and that the fifth even number is 10, so using the for loop is a better prospect.

In addition, Python's range can pass a third argument to create a skip sequence.

With these, you can write below.

for n in range(2,10+1,2):
    print(n)

Unlike this problem, if the conditions are a little more difficult and you don't know when they will be met, it would be easier to write using the while loop as shown in айка <'s answer.