To burp cooling system fill it to the seam in the tank and start it up with the heat on. Let it run until it comes up to temp. You may have to fill it along the way. Don't over fill it tho because coolant expands when it gets hot. Once the car has heat and coolant temp reaches normal temp for a bit you can shut it down and cap it. I usually let them sit a while with cap off and top off as needed. You can always do that step next day if needed. Just don't open it hot. Never reuse a squeeze clamp while changing hoses.
I'm a nut when it comes to newing stuff out before a trip. I guess that's what years of seeing failures does to you. I just had a customer with 07 Silverado take a 7500 mile trip flawlessly towing a camper. He brought it to me before he went.
My recommendations: like said by other posts, fluids and that means all of them. Water pump, tstat, belts, hoses, t's. Check the steering and suspension. Meaning ball joints , control arms, wheel bearings, tie rod ends, sway bar links, bushings. Tune up. Plugs, wires and air filter. Look at the underside. Fuel lines, brake lines, fuel tank straps. Take a peak at the radiator. I had 3 in a week that developed small cracks in the plastic tanks. I'll post a link to my truck I think it's in the last pic. Check the brake pad and rotor condition. Tranny lines, oil cooler lines. Do all the lights work. Make sure that spare comes down and has no dry rot. Check for exhaust leaks. Broken manifold bolts. Another thing is to check the Vlom bolts. I have had a ton of these in with loose Vlom bolts. Check your parking brake. While belts are off spin all the pulleys and make sure no noises come from them. Should be smooth and quiet. couldn't hurt to buy a code reader and a spare coil to take with you as well. Here are some pics of mine. Scroll down a ways and you'll see the rad.
https://photos.app.goo.gl/11ZxEvt7nnnm4N