Do I need to put anything on the seal before I put it in? Like sealant or anything?
No, but the shaft needs to be smooth with no rough places to damage the rubber interior of the seal.
Clean the shaft real good and use some of the leaking oil (clean) to help the seal slip down the shaft.
Do I just pry it out and pound the new one in?
Basically yes. Some are easy to nudge out and some are hard. Some you have to destroy the rubber inside which has a spring inside.
Buy your seal first if you have never done one, so you can see how it's made inside.
Do I need a special tool to get it in?
They do make seal installation tools. These are metal rings that are the same OD as the seal and the ID that slides over the shaft.
Here is an example, but probably NOT what you need.
10 Piece Bearing Race and Seal Driver Set
The concept is that the seal needs to be installed STRAIGHT into the hole. It is a few thousands larger that the hole so it's somewhat pressed in. The housing may be aluminum, and if it gets crooked, this can damage the housing or the new seal.
You can do it by tapping around the edge with a hammer, 360 degrees from side to side, but if you can find an object to slid over the shaft to tap on it's better.
Below is an axle seal installation on av LX188 mower; yours probably looks similar.
This is what it looks like with the seal OUT. (Click to enlarge)
When you take the old one out be prepared for oil to run out.
So clean the shaft up with a file or sandpaper, or whatever
before you remove the seal.
I am not sure how your axle is made. Just remove the wheel and see if you can see the leaking seal.