Just to point out here.. Solenoid and starter relay are two completely different things!
Solenoid A.K.A Starter Solenoid is the roundish piece with a metal casing.
Starter relay is similar looking as the black flasher relay but has 4 prongs (two rows x two columns).
The most "complex" situation is the dead battery could be caused by a bad alternator (A,K.A) Stator, which kills the regulator/rectifier which then causes the battery to boil and die thus causing your starting symptoms.
Now, First things first you have to do is get a Multimeter, one with sound continuity test is most suitable!
Things to test:
-On the engine's left handside, behind the left seat side cover comes there is a plug with three white cables.
Disconnect the plug, set the multimeter on the continuity test and test continuity between all three cables. cable 1 and 2, 1 and 3, 2 and 3. the order doesn't matter, you just have to check all of them. There has to be continuity between all three cables and there CAN NOT BE continuity to ground!
3-phase Alternator Stator Charging System testing with a DVOM meter on a motorcycle Road King - YouTube
- On the same left side of the bike, with the left seat side cover off, you have the airbox!
If you look closely, there is an electric plastic part attached to the airbox with a rubber band, that is the starter relay! Very similar to this one:
To check it you have to set the Multimeter to continuity test!
There are 4 prongs! (some people call them pins)
This time the order of the testing probes IS IMPORTANT!
Test the all the prongs, put the Red testing probe on one prong and check all other three prongs with the black testing probe. Choose another prong and repeat! do it to all of them!
Now, REVERSE, put the black testing probe on one prong and check all the other three prongs with the red testing probe.. Choose another prong and repeat! do it to all of them!
On the Multimeter display, you should see a voltage drop of around 0.5volts. Three prongs should show a voltage drop while one of them should not!
I don't remember which way the testing probes should show the voltage drop, maybe someone else could fill in on this part.
-When the new battery you bought arrives, fit it to the bike and start it!
Get the multimeter, set it to DC 20volts and check how much voltage is going to your battery. Place the red testing probe on the battery positive and the black testing probe on the battery negative!
Let the engine warm up for a minute or two,
Give it a couple of throttles a few times, and check the multimeter display! It should show a voltage of ~13.4 to ~14.7 volts.
Assuming the Alternator (stator) is good, A voltage any lower or any higher than this, means that your regulator/rectifier is damaged and needs to be replaced or it will kill your new battery in no time flat!
One good thing you can and should do is check the battery cables and where they connect to!
Specially the Black ground thick cable that attaches to one of the bolts that holds the e-start motor on top of the engine!
Remove that bolt, it should be full of crap, clean it or replace it or just change it with the other bolt that holds the e-start motor.
Lastly, before that trip you are planning, regrease the wheel bearings, put fresh grease on the lubricating nipples on the swingarm and rear shock! Check your brake pads and properly inflate your tires..
One last thing, good luck and be safe on the trip!