Human Revolution is a prequel to the first Deus Ex so you won't miss out on any preceding storyline by playing HR first. However, HR is built on the same concepts (sneaking, finding alternate routes, talking your way out of problems) as the original and you might feel a little behind the curve since you're picking up the game concepts for the first time. And yes, you may miss some inside jokes about crawling around in ventilation ducts (it happens a lot in the first game, and a lot in HR as well).
Although I haven't finished HR yet, I expect it will tie into the events of Deus Ex (already there are some references to the future of nano-augmentation, a heavy plot point in the original) but you shouldn't miss out on these things, they'll just make sense when you play the original later on.
As for Invisible War, it takes place even later after the events of the original so you won't be missing out on any plot points there. In all honesty, some people don't really consider IW to be a proper sequel as it dropped or reduced a number of features that many people enjoyed from the original (most of which are back in HR). So don't feel bad if you don't play it, ever.
Fortify can trigger a network trace, same as hacking a node, so be very careful when using it. Nuke and Slow are nice: using nuke on a "high level" node is useful (nuke will never trigger a network trace), and using slow is nice when you've set off the trace, and are still trying for the capture.
Some general advice:
First thing to remember: if you get unlucky and set off the alarm early, just disconnect. It'll eat up one hack attempt, but that's much better than getting caught.
Second thing: clear the path to the final node first, before you try to take any datastores. That way, you can jump and attack the final node immediately upon a trace starting. It also increases the effectiveness of slow worms, because you can throw a slow worm, finish hacking a datastore, and often still have time to hack the final node.
Third thing: try to avoid high level nodes. You're far more likely to trigger a trace if you hack them, so save them until last, and consider nuking them if you can't hold off on them. This includes API nodes. It's tempting to get them early, but if you set off an actual network trace while trying to hack an API to slow down a hypothetical network trace...You'll look silly.
Fourth and final piece of advice: if you plan on doing a significant amount of hacking, improve your hacking stealth implants. There is no cheaper, more effective way of improving your skill. Even "skill 1" locks can have a dozen or more nodes, and thats a long time to have to stay lucky. Hacking stealth will lower the chances of starting a trace on every node, and that will let you go a lot longer before the trace starts.
Notes on APIs:
Spam: Spam is nice. It lowers the level of the security scan, which increases the time the trace will take hacking through nodes to your entry point. If you're stuck with a lot of high level nodes, it can be worth it to grab Spam before capturing the risky nodes.
Transfer: This one can be annoying. Whenever you hack it (and they're not usually optional) one node becomes easier to hack and one node becomes more difficult. I had a hack attempt once where one of the primary nodes I needed was behind two transfers, and ended up as a level NINE node, while two nodes I didn't give a damn about got reduced. If there ever were a textbook case for using a Nuke, this was it.
Privilege/Authorization: This API usually makes the registry or the datastores easier to hack. If you have more than one datastore, it's almost always worth it to hack this API.
Best Answer
It really doesn't matter which one you pick, for Deus Ex. It might matter if they bring out another game that's a prequel to Deus Ex.
Really the only 3 things that tie into Deus Ex, you as the player had no control over: