Phrase for overusing just-learned skills?

Solution 1:

You might consider overzealous adopter.

To adopt is to take up and practice or use. An adopter is one who adopts.

To be zealous is to have an enthusiastic commitment to. Add "over" and the sense is that the commitment is extreme or beyond what is called for.

Solution 2:

To paraphrase Maslow's Law...

Got a hammer, now everything looks like a nail!

The "standard" version is usually given as If all you have is a hammer, everything looks like a nail, but in practice people often use it in contexts where said hammer has only recently been acquired.

Solution 3:

You could say someone who does this is like a kid with a new toy. It often just means someone is particularly pleased, but sometimes includes the connotation of overuse.

Solution 4:

Not necessarily relating to a new skill, but "flavour of the month" might be in the ball park if used in the right way:

He only learned how to use the screwdriver yesterday, but now its the flavour of the month.