1986 Honda fourtrax 250 starts with solenoid olny

Well, first you'll need to replace that solenoid unless you bought an OEM Honda solenoid. Chinies knockoff parts are garbage. If your solenoid is OEM Honda, make sure the trans is in neutral, the neutral light should come on. If you don't get a neutral light you'll have to diagnose that issue. Is your new solenoid a genuine Honda part?

Welcome to the forums!

Great! You can click this LINK to download a copy of the service manual. Check the wiring harnesses out making sure they are original, in good condition, not spliced on or hacked up etc. Replace any that are... Make sure everything is plugged into the harnesses and those connections are clean and tight.

Then check the fuses in the fuse box for any blown, corroded or loose fitting fuses. Finally, check the ignition switch, the neutral switch, the kill switch and the starter button switch. All of those are covered in the manual. All are very easy to check too, just follow directions. You can use a multimeter or a test light, whichever you have handy.

Have fun!