I've only done the french drain thing; I have no idea what would be required with tying into a storm sewer; you'd likely want to contact the local government and/or water company to ask.
Anytime you're digging, you'll want to be aware of where underground utilities are. In the DC area, you call 'Miss Utility', and they'll mark out everything. (I think technically, you're not required if you're digging by hand, but it's still a good idea).
You might also want to take a look at what large trees are nearby, as you might have to deal with roots -- you can cut them as you go, but that might end up killing the tree or weakening it so it sheds branches, and large branches falling on your house or car is not a good thing. (especially when you have no power, so get trapped in your (damaged) home).
You'll want to look at what's down-slope of the area you're dealing with, and how much area you have to absorb the water once you move it down there, if you're going the french drain route. If it's an area that already gets pretty marshy after a rain, it's not going to absorb anything, so you'll need to go far enough out 'til you find somewhere that'll take the water. You also don't want to try to force it out near the property line, or you might piss off your neighbors (and if they're lawyers, or just happen to like suing, it could get messy).
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And after all of that, you might want to look into if the area's abnormally wet -- one of my friends had problems for years, and dug up his yard multiple times, trying to get everything fixed, redid his basement at least twice because of water damage. And it turned out that there was a broken water pipe next to his house that was saturating the lawn, so any significant rain couldn't be absorbed.
Your basic plan sounds workable. I assume your crawl space is adjacent to a full basement area? I have a couple of questions. Where does the existing drain tile system empty? Is there a difference in level between existing basement floor and footings of the crawl space? the answers to these questions may change my answer to you. But basically, adding a French drain or perimeter drain tile should help your situation. After your trench is dug, be sure to line it with several inches of gravel/crushed stone. Use perforated pipe with inlet holes or slots on only one half of the surface and be sure the slots are facing up when installed. Wrap the pipe in common landscape cloth before backfilling with crushed stone to prevent the drain pipe from filling with sediment. Lay another layer of cloth over the stone and complete the backfill with clean grave. Not sure how you plan to connect to the basement drain system, but that is another topic. This kind of work is always difficult, especially when working in a crawl space with limited head room. Good luck.
Best Answer
You can do the drain system and gutters/downspouts in either order. Assuming you are installing an in-ground drain system, you'll need to use the appropriate fittings to receive water from downspouts. You will want to install these when you're first burying the drain system, so you don't have to dig it up later to add them.
If you do gutters/downspouts first, you'll just make sure to route your drains so that you have intakes at the right spots. And you'll likely modify the ends of the downspouts slightly so they go straight down into the right spot.
If you do the drain system first, you'll have to place your downspouts such that they can flow into the drain intakes. Also pretty easy to do at installation time.
No matter what order you go, you should have a design for the whole system in mind before you start. The shape of your building and site may constrain both the gutter/downspout system and the drain system in different ways. For example, you may need a certain minimum number of downspouts to handle the expected rain flow.