Welcome again haha! Nice of you to drop by   
Good to know that you have read around before asking~

Like what the other sifus have mentioned, the 18-55mm focal range is one of the most versatile and most often used, so its a great lens to being around for your trips, especially so since its so small and light The 18-105mm/140mm kitlens are great too, offering more zoom- suitable for zoo, bird watching, lengloi spying etc., but of course its heavier and more expensive la haha

Usually for starters, its best to just start from mode P > S > A and finally M. The learning curve is extremely steep if you start straight with M mode, and if you are not determined enough, you might just become so demoralized that you stop photography altogether (some of my friends went through that ), the idea is to take it slow and enjoying the process
Each mode will let you know how and when to adjust the settings i.e. ISO/Shutter speed/aperture, and their link to one another to get the final photo you want 

Which one to use depends on your level of skills now, you can't always shoot at P mode since there's really not much for you to adjust haha If you already know the basics, then you might want to try out Aperture mode. Its pretty simple as long as you are shooting outdoors and in the sun~

Learn to use higher F values (smaller aperture) for scenery shots (or scenery shots with a person in front) e.g. F8-16 for instance: so that everything in the shot will be in focus

Learn to use lower F values (larger aperture) for portraits etc. so that the main subject is in focus, while the background becomes blurred out: to attract the viewer to look at your subject


* Shots were taken during my Taiwan with an 24-70mm, which is sort of equivalent to the 18-55mm on a DX camera like your D5200 
* Oh ya, I usually shoot with Aperture mode too haha XD I only use manual when I shoot macro, or want to shoot long exposure shots