There is no real story to the first Half-life. Definitely nothing that particularly happens while you play. If all you want to know is the plot, I'd recommend just reading about it rather than actually playing through, as it's a pretty damn long game. It was made back in a time when people made FPS games that actually had some length to them.
You're Gordon Freeman, you work in the mysterious, suspiciously well-off Black Mesa research centre in New Mexico. During one day of performing routine - yet strangely, never properly explained - experiments involving giant rotating lasers, stupidly positioned switches and pushing weird mini-tractors in an obviously unsafe environment, you're given an unusual "sample" to use in them. When you push the weird little rock into the big glowing beam, all hell breaks loose and aliens begin invading. Hearing about this terrible breach of inter-dimensional ettiquette, the government wisely decides that letting weird creatures that fire lightning bolts and parasitic headcrabs roam free just isn't the done thing, and dispatches some friendly army types to point bullets at anything that moves - including you. But then, deciding that this just isn't enough, they send some
more army types (these ones are in black, that's how you know they're OOOOH EVIL) to kill everyone again, and the first army types to boot. And
then, because nothing says bureaucracy like doing things in triplicate, they decide to be done with the whole mess, and just nuke the entire place. Before they manage that though, you (with a little help from three scientists and one security guard - who apparently cover
every shift in the entire centre) wander through the obscenely large Mesa complex, go on a bit of a train ride, launch a rocket-borne satellite, blow up a handful of helicopters/tanks, order a couple of airstrikes, kill all manner of weird-looking beasties, and generally ponce about in your swanky hazard suit. Then you mince off to an alien planet for a bit, leap from floating rock to floating rock, fight a giant alien testicle, and eventually find yourself shooting a giant flying baby in the head with a large amount of rockets, only for your closure to involve getting chatted up by a suited "man" who is more cryptic and has more unanswered questions over his head than an entire series of Lost. Fade to black.
Half life 1 is SO much harder than Half life 2... you heard it here first.
EDIT: For some reason I thought this said "better" last night. Your facty status is revoked, good sir!