I know exactly what is wrong here, its actually very common with this car infact i just repaired the EXACT same problem for 10 bucks i'm going to tell you how to fix your problem. Its your
neutral safety switch.
I'm going to assume that when you try to start to car you the dash light and radio goes off and comes back on when your key is back at the ACC position. Sometimes is might act like its starting to turn over and then won't and other times complete silence.
The problem you are experiencing is dirty contacts in your neutral safety switch. These switches actually never go bad per say however after 20 years of use one of two things generally starts to happen. You either build up a ton of dirt in it or it comes out of alignment.
If you have never cleaned the switch it would be a good time to go ahead and do it.
Instead of switching gears push the gear shifter hard into park. If the car starts after pushing it up into park at the same time but when no pressure is applied to the gear shifter then its full of gunk. If it starts after not applying pressure then it just needs adjustment which is easy as long as you have a 12 or 10 mm socket. Can't remember which at the moment.
Go to you local auto parts store and get the following,
1. Metric socket set if you don't' have one.
2. I can of electric parts cleaner
3. 1 small pack of diaelectric grease/bulb grease
4. Phillips head screw driver
5. Automotive pick set. Got mine at kmart for $1.00! WOOOT
First lets locate the switch. While standing in front of the car with the hood up look to the lower right side of radiator down below the battery. You will see the switch on the bell housing. It is V shaped. Mine was filthy and blended in really well.
Disconnect the connector to the neutral safety switch and remove the nut of the front of it. Depending on your model you might have to put the car in neutral to get it off or leave it park.
Remove the switch. You will need a phillips head screwdriver. it is held together with 4-5 screws. Take the screws out you will see a little cheap-o gasket around remove and set aside. Carefully remove the contact arm. It has two contact blades in it that are spring loaded BE CAREFUL if you loose these contacts your SOL.
Use a can of electrical parts cleaner on it switch and clean out the goop. If you have an automotive pick set it will really come inhandy with removing the old dielectric grease.
Get a small pack of dieletric grease/ bulb grease and greasu up the switch. Be sure to clean the contacts and lightly sand them with a nail file so they will make good contact. I removed the springs and stretched them out a bit, after 20 years making them 2mm longer isn't going to hurt just be careful not to over stretch. Grease everything in the switch all contacts, hell everything.
Put it back together and slap it back on and tighten it down.
Now here is the trick and you will need a bid after it is back on you might experience the same problem however if you have a friend that can push the gear shifter forward you can see how the switch needs to be adjusted and make adjustments accordingly. If you take your time you can have this done in a hour.
The Good, The Bad and The Ugly
Now lets say the switch is horribly damaged i.e cracked by a rock or something. You can try to weld it back together or just follow the link below you should replace it, or if you just want replace it you can.
A problem that I've faced is that rebuilds of this switch can last for a few years or a few months. It really depends on your mechanical knowledge.
Be prepared to get dirty. Your best way to access this is not from above but from blow after removing the splash guard on the bottom of the engine.
Click here for replacement part