Guitar hardware:
For recording:
The Stratocaster and the Les Paul are both extremely versatile instruments that have been used to create all sorts of sounds outside of their original design parameters. It's useful to have at least one of each in your sonic arsenal. The Stratocaster especially is an engineering masterpiece, a work of art. On the non-deluxe models, fitting locking tuners like Schallers, and replacing the trem block and bridge assembly with Callaham models is recommended. The Callaham service is outstanding. You may want to replace the stock pups too.
You'll probably want a capo. The Kyser Quick-Change spring grip capo is very good, though it's also quite a strong spring which means it can eat your fret wire if used long term. The G7 capo with the soft clutch mechanism is a reasonable contemporary alternative that may treat your fret wire with more care.
Strings don't seem to matter a lot, but you'll probably be able to tell a slight difference in playing. D'Addario strings give a warmer, chunkier feel than Ernie Ball, though Ernie Balls are the most popular around. As for gauge, try to train your fingers to cope with .010, to get those unwound strings sounding less plinky. Flat-rounds are pretty cool, but they tend to feel a bit damp with time. The pure-nickel strings sound brilliant for the first week or so, but then get very muddy thereafter. Good for concerts, at least, or people who don't mind changing their strings one per week.
You can plug a guitar straight into a computer soundcard using a guitar jack to line in port adaptor, but the sound using this method won't be as clean as you can get it. Using a DI will be an incremental improvement. There are many things to look for in a good DI, including number of input ports and dials for various features. The most important of all, though, is latency. If you're recording into a computer, you don't want to hear your guitar half a second after you actually play a note.
The Focusrite Scarlett 2i2 is one of the best performing for latency, and is quite cheap. It can be prone to strange buffering output, though it's difficult to tell whether that is due to the DI or due to buffer problems on the software end. This problem is intermittent, anyway, and usually resolves itself with a "turn it on and off again" cycle or two.
You can couple software amplification with live microphone pickup, even with an electric guitar, for some very nice electro-acoustic effects. Software amp emulation doesn't tend to get the highs right but works very well with the bass. This isn't too surprising because humans tend to be very finely tuned in their aural fidelity in the speech range, which is where the highs on a guitar lie. Using a microphone to provide high quality acoustic reproduction in that range, and then supplementing that with software amp emulation for the lows, works very nicely. The SM57 is an extremely good microphone, with a beautiful flat, clear response curve. The Scarlett 2i2 comes with XLR microphone inputs. You can get reasonable gain from the SM57 by turning the Scarlett's input dial all the way to maximum, but for other DIs you may find you need to buy a pre-amp too which will complicate the toolchain a little.
On the software side, guitar amplification emulators have come a long way. There are several leading products, but AmpliTube is quite reasonable, though very expensive. You should look out for discounts if you want to pick it up. It's possible to make a toolchain out of only free(-as-in-beer) software. Couple the Le456 amp from LePou with some of the free RedWirez IRs for cabinets, and use the Bricasti reverb IRs that are floating about for post-effects, and you'll get a very good sound.
For the actual process of recording, a lot of people use GarageBand on OS X, but REAPER seems much better, though it does cost a little money. But if you compare it to more established DAWs such as Cubase and FL Studio, REAPER seems to be a considerable bargain. Get it whilst it's hot.
There are some compression plugins that come with REAPER which work well. UpStereo is another excellent free plugin if you need to seriously boost your sound. It can feel a bit grainy if you overdo it, though, so keep it compact.