A computer. Speakers/Headphones. At least a DAW and some other software (in the form of plug-ins, if you wish). An audio interface. A microphone (or more). A portable recorder.

Sound/sample libraries, if you're making soundtracks (which are not about and aren't full-on "sound design").

This is entirely dependent on what sort of work you'd plan to do, and what exactly your budget was. Without an example of budget etc it's hard to give specific examples of equipment. But if you really had no money to spend and could get access to a computer then it'd be entirely possible to start creating basic sound effects for use in something like a simple browser game using the voice recorder on a mobile (cell) phone and freeware like Audacity.

Strapped for cash? I'd buy nothing new - secondhand Mac, secondhand MBox & ProTools10 ilok, a shotgun mic and a recorder. You don't need other plugins because if you are starting out there is a LOT to learn just with ProTools... and frankly, if you cannot make interesting sounds with just those four things, then its your imagination & creativity that needs focusing on

Hmm, probably an H4n, NTG3, PT10, RX2, a pair of Yamaha HS50Ms, sony 7506. Those would be my main essentials. RX2 specifically for removing preamp hiss from the H4n. PT10 is fine out of the box nowadays, but Ozone 5 if I had to choose one 3rd party plugin.

H4n - 300 NTG3 - 700 PT10 - 700 (or a cheaper student version) HS50m pair - 300 7506 - 80 RX2 - 300 Ozone 5 Adv - 800

Total = about $3200.. on the high end

Oh, and AudioFinder for 70 bucks is a great buy too! Use it every day

I'd do like I've always done, buy a few vital things with the highest quality possible (it's more expensive in the long run to upgrade), and work only with the sources until I could afford more trustworthy monitoring so I could start filtering and processing more seriously.

Of course I wouldn't be nowhere near buying an Aaton at that time (still isn't), so I'd go for the best thing for the buck - a Fostex FR2Le. I actually work with the full FR2 right now. Then I'd get a Line Audio CM3 cardioid and a Sennheiser MKH416, a pair of BeyerDynamic DT250 headphones, and some Supra-cable. Though very cheap, the CM3 is still a valuable asset to my set.

With that I would record everything I could find 'till I could afford the DAW-part. For the audio interface I'd go for a LynxTWO-card to begin with, Steinberg Nuendo for DAW, and the best monitors I could find for my money. Right now it might very well be a pair of old Alesis Monitor One mkII, though chances are I'd very well find something else I would like if I tried. The M-One are not the easiest to mix in, but they're pretty straight-lined and far from flattering, and when I upgrade they'll make a good addition to much bigger systems to confirm the mix.

But frankly, when I started out most of these things didn't exist. At the very beginning I had a really shitty DV-recorder with auto-gain and a Röde NT3, though my Yamaha DSP Factory-cards and Monitor One (mkII, the first model was more lackluster) was very nice at the time! All of the sounds recorded during those about two years is completely useless today.

I think i would go for a laptop with ableton live suite, apogee duet 2 interface, zoom H4N, Røde NT1A, Sony 7506 and maybe active Event monitors. I think there is more creative possibilities with ableton than PT, if i had to pick one to start with. I don't think we'll ever stop upgrading our gear anyway, so the thing is to get the best of what we can get. Even now after 20 years and a lot of quality gear I'm sure i'll change it at some point. The only thing i keep is good microphones, vintage keys and tube preamps/compressors etc.

Ramblings ahead!

This seems to have focused on what to buy... (sorry I failed to read the question properly, I still think my response is valid so i won't delete it).

Define sound design...

Do you mean working with sound for film/tv etc in general, designing it's soundtrack? Do you mean working with sound for AAA games designing it's soundtrack? Do you mean designing sounds for yourself as art? Do you mean designing sounds for independents to use in ios or online games etc?

All the above could be defined as doing sound design. And the paths to get anywhere is very different depending on where you're goals are.

Yes, designing sounds is a vital knowledge and part of the needed skillset for working for film/TV. But you will never ever get a job starting out as a "sound designer". You  start out assisting on a shoot pulling cables and fetching coffe, or editing basic sound effects/foley/ADR for less demanding projects. Until you have proven that you can find/choose the right sounds get them in sync and complete your tasks within the predefined time all while beeing a nice person to be around. You will not be considered for anything but very basic tasks. The closest you'd get to "sound design" in a SSD way is going out to record some sound effects that's not vital for the story but would be nice to have and if the newbie (you) screws up its no big deal. 

Major games are similar but still very different.

Art, small games and school films (for free), is different still.  These are the areas that pay the least but still require you to know your tools, own your tools, understand the craft. Delivery is what is mostly different, with these kinds of projects time is likely to be less of an issue. Either because  art is expected to take time, school projects understand the learning curve and small/independent games will have a limited amount of assets you need to deliver so you are likely to be able to spend more time on each asset.

What you need in terms of knowledge and preparedness is very different. What you need in terms of tools/equipment is also very different. What it will cost you is also very different. How to get there is also very different.

Now what was the question? 

If I was just starting out, looking to learn about audio, and was strapped for cash I would buy the following:

Cheap Windows laptop (with built in audio): $500 or less

Reaper: $60 for the non-pro license

Sony PCM-M10 Recorder: $250

Decent Headphones: $100

Download/Buy Sound Effects online as needed (freesound.org, Sounddogs, Rabbit Ears, The Recordist, Hiss and a Roar, etc)

That's under $1,000 usd and enough gear to get started learning to record/edit sound. No need to go into huge debt to start up. Once you gain more knowledge/experiece/work, you can start to upgrade the recorder and computer, buy an audio interface, plugins, mics and speakers, and move to other software should you choose.

Film sound:

• A season/membership pass to see (and hear) good films.
• A few good books, notably "The Conversations", "Tuning of the World", "The expressive power of music, voice & sound effects", "Practical Art for Motion Picture Sound".
• Any mac

Game sound:

• A small chunk of cash for carefully selected indy games.
• A good video card.
• Any PC

Both:

• A pair of headphones I can learn really really, really well.
• Any pair of speakers that aren't abysmal.
• A Sony PCM-M10 recorder
• A battery-powered mono condenser/electret microphone
• Adobe Audition