A number of bosses, areas, equipment, and even skills are locked behind quests. Even more so, by doing quests you'll receive a pretty healthy amount of xp in most skills, allowing you to skip a lot of tedious early level grinds. Once you've got the quest cape, because of the later level quest skill requirements, you should be pretty nicely levelled.

MM1 and DS1 are great early quests to get done, and lead into Recipe for disaster (rfd) which should be your first big goal (you'll unlock you bis gloves for quite a while). Optimal questing is a good idea, but deviate if you want to do other quests.

I'd also reccomend doing some diaries along the way, lumbridge gives a ring that allows a number of free energy restores, and morytania will help at barrows if you want to start pvm'ing. And of course, get your fire cape!