Recently, my PSP started to not charge properly. I quickly found out it was what plugs into the PSP, and not the wall.

I tried messing around with it, but at most it would charge for a few seconds then not charge, then charge again and so on. I tried checking the battery and such but it wasn't bad. Then, the thing refused to turn on anymore. The battery wasn't even dead yet, yet the system wouldn't even turn on. Then, randomly, it came on without me even touching the power switch.

Now, it won't charge or turn on.

Am I going to have to replace anything?

Most likely the connection point in your PSP has some corrosion, it happens over time with any electronic device. if you're comfortable with electronics you could probably take it apart, and simply clean the connection point.

Edit: They also have charger bases for PSP batteries, so you could go that route as well.

I've had it for a month.

Oh well, if you bought it new it's probably still under warranty, so it could just be a bad part, or something else. Generally speaking though, when you have trouble with a charger it's usually that you're just simply not getting a connection with the plug due to come corrosion.

Do you have any suggestions to fix it?

Knowing that's it's new means it could be some other problem and you should probably give Sony's customer support a call and see what they say.