I own a 1997 Eclipse RS 110,000 miles on it, recently i ran it low of trans fluid. So i filled it back up, now the car won't shift out of first gear when its warm. Please help me with this problem thank you

Auto or manual?

automatic :you may have burnt out the electronic shifting solenoids in the box

best take it to an automatic tranny repairer to have it checked out

Is there anyway me and my dad would be able to replace them ourselves or is a auto transmission difficult to work on?

wouldn't shift = auto transmission diying

auto's are finicky about adding or flushing fluid. sometimes there is a buildup of gunk, dirt or shavings that can dislodge and clog up one of the valve channels. Since it's already having problems try having it flushed, with a new filter and have the selector removed and cleaned. if this doesn't work then it's time for the trans shop

Okay thanks i will do that.

If its an automatic transmission, how would you know that its not shifting to the first gear?

Generally if there is a problem with an automatic trans, they go into what is called "safe mode".
This is when the trans permanently goes into 1st or second gear, and stays there. This a safety phase, so that you won't destroy the trans.
Drop the pan, and look at the fluid, and smell it. If it is black, or very dark, or smells like rotten a$$, or both, you most likely fried something.
Dump the oil out of the pan, but leave the sediment in the pan. Look for clutch material, or metal shavings. This is usuall very bad, and means a new tranny.
If that stuff isn't present you might be in luck.
Drop/remove the valve body, and very carefully remove solenoids/solenoid packs (watch for check balls, and make sure to keep them, and note where they came fromso you ca put them back when you reassemble the valve body).
Spray everything down with carb cleaner, making sure to clean everything well.
Reassemble, fill tranny with appropriate fluid (probably dexron/mercon), and statrt engine, in neutral, not park. Run for a few minutes, and then check fluid level, fill if needed. Make sure you stay in neutral this whole time. Import trannys circulate fluid best in neutral, unlike domestics, which need to be in park.
When you have completed these steps, and things are still fine, take it out for a drive, and go easy on it for the first hour or so of driving, assumeing it is now shiffting. During this time the trans is "re-learning" how to shift.
If tings do not go well, buy a new trans. But, a waning, anew trans at a trans shop, is really expensive. Place I used to work at charge about $3500, because parts were expensive, and sometimes hard to find, since they were mostly special order.
IMO, if it needs a new trans, sell the car, and use the money to buy a new eclipse/talon. Stick with a manual this time if you can.
Best of luck dude!