I have this (IMHO) cool idea for a general storyline I want to write about, and one or two ideas for characters. However, I'm really struggling to go about fleshing it out. It's like having a string for a necklace with two big beads at the ends and one or two minor beads along the way. Are there any techniques for fleshing out a story? Coming up with new elements that can be used to join the beginning and end together, given that you have this vague overall idea for a storyline?
It sounds as though you have two pieces of the story right now: the premise and the conclusion. In order to come up with a complete plot you will want to come up with one or more developments, plot points that change the direction of the story (in a mystery they would be called "plot twists"). If you're using the common three act structure you will want three of these. So, Act I would begin with the premise and end with the first development, Act II would go from there to the second development, and Act III would go on to the conclusion.
Depending on the length you want to work at, each act will need to have some internal structure of its own.
As far as determining what these developments should be, a good question to ask yourself is where is the conflict? It could be something as straightforward as saving the world from evil, or as complex as the conflicts inside a family having issues. Generally there will be smaller conflicts that also play out. The Fellowship of the Ring sets out to save Middle Earth, but Boromir feels that the Ring should be used to save Gondor etc.
Adding an internal conflict can help create tension. The hero has to stop Hitler from getting an atomic bomb but also his father think's of him as a failure.
Basically, play around with the plot and characters you have like they're LEGO pieces, and assemble them in different configurations. Then bust 'em up and see what you can put together from the results.
Somebody has said (probably on this site) that a plot goes like this:
He wanted [goal]. So he [action]. But then [conflict], which caused [tension]. So he [action], and [resolution].
Repeat this a bunch of times, and you've got a plot. Of course, if you don't disguise it better than that, your story will be awful.
Also, along the way you'll want [characterization], to help explain the [goals], [actions], [conflicts], and [tensions] of your characters. Plus, you'll need some sort of [setting] for your storyline. (Remember too, that the [characterization] and [setting] in turn can often shape the storyline.)
If you fill in the brackets well, you'll have a fine story.