I know when I was (much much) younger and read <i>The Lord of the Rings</i> and Anne McCaffrey's Pern novels I wanted to visit those worlds. And indeed there is a solid history of fantasy novels featuring characters from 'our' world crossing to the fantasy realm and in some cases the opposite.
Now on the whole I'm not a fan of this plot device; it seems in some ways to break the concept of a separate world and any explanations of how it works are always a bit hand wavy. I can see the appeal - not only do you get someone with our point of view suddenly thrust into a world of strange races and magic but there can be some fish out of water and culture clash humour.
There are notable examples through literature of course. <i>Alice's Adventures in Wonderland</i> and <i>Through the Looking Glass, and What Alice Found There</i> both feature the heroine travelling to fantastical places seemingly just a rabbit hole or mirror away from ours, but populated by strange characters and possibilities. Alice takes it all in her stride, announcing how peculiar everything is. However these are explained in the book as merely being, in effect, dreams and not really crossing from one world to another.
The Narnia books by CS Lewis are another obvious example. Here the first mechanism is the wardrobe which four children discover contains a gateway to a world of talking animals. What is neat here is that the wardrobe is explained to have been made from a tree which grew from seeds from Narnia so there is some logic there. But as the series continues it seems that crossing to Narnia can happen a variety of ways and it's never really explained. It's also clear that the wardrobe (at least) is not a simple gateway; the children go through in their normal clothes and age to adulthood in Narnia. When they return they are children again and dressed in those same clothes. How did that happen? As a child reading the stories I don't think it bothered me but looking back on it I find that aspect the elephant in the room. If the wardrobe is a portal to Narnia it should work both ways to my mind. It just seems a little too pat and prosaic.
The <i>Truth Teller</i> series by Kurt Chambers has a superficially similar mechanism where the heroine, Charlotte, is transported to a land of elves and dwarves by use of a magic talisman. For me this actually worked well since it was constructed and designed (magically one presumes) to allow her passage as she is the 'chosen one' (fantasy loves a good prophesy!) therefore any side effects, such as time not having passed in the 'real world' during her adventures can be explained as a deliberate intention. Travel is also possible in the other direction for both friends and foes via a similar pendant.
Neil Gaiman's <i>Neverwhere</i> and China Mieville's <i>Un Lun Dun</i> have a slightly different concept, that our cities have an alternative version just a shadow away and that what happens in our version is reflected in a strange and fragmented way in the other world. These worlds are similar to Wonderland in that they essentially co-exist with our world and are accessible via particular channels. They are not really separate worlds but symbiotic with our own. In fact both Gaiman and Mieville typically tackle this the other way around, with the fantasy elements being present in our own world rather than having to invent a new one.
In <i>The Pilgrims</i> Will Elliott uses a simple and completely unexplained method of access to another world - a door set into the wall of an underpass. This enables the two protagonists from our world to enter the other but it's not really explained why this happens or who put the door there. Since presumably these worlds have been linked for some time, what was there before red wooden doors were common? As a quick and dirty mechanism to allow characters to step between worlds but for it does somewhat jar against the rest of the narrative.
What first got me thinking about this years and years ago (and make be pretty jaded about the whole our world heroes travelling to fantasy worlds) was <i>The Summer Tree</i> by Guy Gavriel Kay. For some reason the whole premise (which I have now forgotten) just didn't sit at all right when I read the book and just seemed to be more wish fulfillment than a solid premise for a fantasy work. There is no slight intended and I know many people enjoy his work but for me it didn't gel.
Much as my much younger self might have imagined travelling to the worlds he read about, it is certain that actually writing a story with that as the basis is bound to cause problems with the reader unless treated with the utmost care and that most of the time this is actually achieved with a little sleight of hand (or word), creating the fantasy world so that it is really a part of ours rather than somewhere separate that must be travelled to. Sometimes a little wish fulfillment is a good thing but must be handled with care.