I just bought an 05 Neon
 

Recommend me things I can do to help improve gas mileage.

Anything from $0-$500 MAX.

Don't drive over 65 mph
Pump the tires up to 90% of the max pressure on the sidewall.
Fill the trans-axle with a good synthetic such as Mobil 1 or Redline.
Look at grill blocking.

Make sure the car is up to spec.
Alignment is right on specs
Fresh spark plugs and filters
Make sure you have no brake drag.
Brake drag is when the pads are just
touching the rotors or drums producing drag.

If its a manual, find a 95 or 96 base model neon gear box. These have much taller gearing in 4th and 5th. Check out neons.org. Lots of info there.

Pump the tires up to 115-120% of max sidewall. Measure in a.m. when cold before driving.

All the other stuff written above, I second that stuff. I dunno about the taller gearing in 95-96 but my guess is that tjts1 knows his stuff.

No, he/she doesn't know their stuff.

https://www.neons.org/tire/speed.shtml

You can plainly see the ratios of the different transmissions neons are equipped with.

If you have the base Neon you already have the optimum set up.
2003-2005 Base SOHC
3.55 final drive with .72 OD

I have a 2000 Neon with an automatic. Pumping up the tires helped me the most. I put in a grill block today. I'll let you know how that helps. Neons have great potential. They are pretty aerodynamic and underrated by the epa for mpg's. I drive mostly highway and get 34mpg. If you have an auto, stick with the fluid that's in it. It is really good from the factory and you can only get it from a chrysler dealer. It also cost $6.95 a quart. I had to buy some recently. Not mpg related but remove the big rubber weatherstripping against the firewall under the hood. Dyno proven 1 hp gain. Just a tid bit for you.