97 Dodge Ram 1500 4x4: headlights and tail lights won't come on together at same time

These are all different circuits. The brake lights are on a 10amp fuse - the headlights are on a 50amp fuse - and the parking and dash lights are on a 20amp fuse.
Do you have power going into the headlight switch and the brake light switch ?

Check the headlight switch. The power goes from the fuse to the headlight switch and then to the dash and running lights.

usually when the headlights goes off so does the tail lights thats mean the brake light switch but the side markers are staying on too. it can't be the fuse since they still light. the only thing that breaks the currant is the switch. it sounds like there,s a problem with the headlight switch. the headlight switch operates the side markers and the tail lights along with the head lights. Hope this can helper .

The parking lights would not normally come on unless you turn them on with the headlight switch. The daytime running lights would come on in drive and go off in park.
If you can't turn the parking lights on with the headlight switch, you would check the fuse, the relay if equipped, and the headlight switch.

First check the fuse, the clearance lights are not factory in most cases and the wiring gets shorted from poor wiring. Once you replace the fuse see if the lights come on, if the fuse blows right away find the wiring to the clearance lights and disconnect them, if the short goes away it is a I said the wiring is shorted. If you don't have the owner's manual you should be able to download it at the link provided. It should be fuse #10 a 20 AMP in the interior fuse box. http://www.fordservicecontent.com/Ford_Content/catalog/owner_guides/97ecoog1e.pdf

I had a problem with my 95 Dakota with the headlights(they would go out intermittently,REAL fun on a pitch black stretch of road!).I found that wiggling the knob made them come back on.I bought a new switch,but when I went to replace it,I found one of the connector terminals was loose,and it melted the connector body.Since the switch had common sized male quick-disconnect type terminals,I just drilled out the body and used a female quick disconnect to repair it(you won't see it after putting the dash back together anyway).Check the connector,do a "drag test on the connector pins if you can't see anything obvious(use a male terminal pin to see if any female connector terminals feel loose).

Replace the headlight switch, it's not uncommon for the switch to go bad and still operate 1 or 2 other items.

Here's a link to a pdf file containing the headlight wiring diagram for your year model Dodge Ram 1500..
I can't help but notice that you say you've replaced the headlight relay? That's quite odd, because there is NO headlight relay on this vehicle- only a 40 amp fuse located in the power distribution box that sits behind the battery inside the engine bay.
The switch is notorious for burning out because there is NO RELAY on this vehicle, and there is a recall (#819) but this only affects 3500 models even though the headlight wiring is the same for 1500,2500 and 3500 model Rams.
Here are two more links that will help you troubleshoot your headlights problems.
HereInformation on Putting a Relay in the Wiring System