Hi JuneBug - welcome to the Forum!
You ask a very good question - this is an issue that is often misunderstood, especially by people new to travel writing who think that the moment they decide to be a travel writer, all their travel will be free from then on. Not true.
Even Travel Writers who have been published for years and have made a "name" for themselves to still pay for (some even if not all) of their travel expenses.
Generally speaking you do have to have some published articles under your belt - whether that is online or offline (print), however if it's online it is better if it is for reputable websites with a decently significant amount of traffic (visitors). This makes it easier for you to get invitations for what is called "Media Trips" - when a destination sponsors a trip for writers to do just what you ask about - write a travel article about them that will get published.
But that shouldn't stop you from approaching a destination to ask them if they are open to what you propose. My suggestion would be to start where you live - first, write a few articles about local resorts or attractions and submit them to your local newspapers. Once you have something published, you'll have more clout when you contact a local hotel or resort and speak to the person in charge of their PR department about doing an article about them in exchange for a free weekend.
Start small and work your way up to trips elsewhere. As our own site grows, we'll be adding a section for Media Trips also, so stay tuned! And good luck!