ill guess this is your first mmo judging by the question, but that isnt bad at all. i had the same thoughts when i first started playing.
first off dont think of you playing a game on your computer. you are playing a character who is accessing a world on a server up in boston. so everything you do is done so there.
so if you simply log out on your client your character is actually leaving that space on the server. if you log back in your character will be in the same spot as you left with everything just the way you left it.
EXCEPT if you are in combat or died do to misadventure, aka falling from a high place. if you are in combat and you log off your character is still in game fighting, or dying rather since you arent there to play. so if "it" hits the fan and you log off in combat the next time you log on your character will be in the place you would have been if you simply took a step back away from your computer and let your character fend for itself. in the rez circle.
so some quick points of advice.
log out in a safe area. if you are able to walk away for 5 minutes and come back and find yourself not killed by some roaming bear or whatever then its a safe place.
take your time, there are no "do overs". imagine you are playing the game as it is your life. kinda dramatic but its the way it is. you are driving down the road and run into a tree you cant simply reload your last save point. just like if you are running in the old forest and get a bunch of trees and wolves and spiders chasing you cant simply stop and reload your last save point. you are living in the moment.
and above all. just enjoy the game . in the end dying doenst matter. its how you learn. and ive learned a lot
I think I've heard about your problem before. Normally it's as people have written before. You normally don't need to save, and can't do so.
But in the very first little area in game. Before even the starter area I belive it doesn't save you properly. So you may need to go through that first little bit, sbout 1 lvl, and come out in the bigger starter area for the real saving to begin.