Someone just wrote to me asking for advice on getting started building a dapper wardrobe, and of course I have LOTS of advice. I figured I'd share it here too.
Style inspiration: I love looking at portraits of the original dapper dandy, Beau Brummel, and his fellow Regency-era men. A quick Google search for "Regency men's clothing" will give you a ton. If you like more flamboyant styles, researching the fops and fribbles who predated him is also a lot of fun. In the modern era, try looking up drag kings, who have fabulous style and do interesting things with the basics of menswear. Or flip through the catalogs/websites of high-end butchwear companies even though you can't afford anything they make.
Buying clothes: EBAY EBAY EBAY. I'm a boy's size 16/18 in shirts, and it turns out there are a lot of teen boys who wear very nice shirts once and outgrow them, and then their parents put them (the shirts, not the boys) up on Ebay for pennies. The best thing is that many of them are posted with measurements as well as size numbers. Buy a few different brands and see what fits. You can also go to a big department store and try on things from the boys section. Men's trousers are more difficult to find unless you're skinny all the way down; I have a small waist and a big butt, and I do pretty well with Old Navy men's jeans as long as they're a tapered fit. Don't shy away from fitted clothing. Baggy pants won't hide your shape--they'll just make you look shlumpy, and you won't feel good about yourself, which hampers the self-confident attitude that's the true hallmark of the dandy.
In one word, I teach you the secret of success in dapperness: ACCESSORIZE.
The most obvious accessory is the tie. If you have a short torso, regular men's ties will be much too long for you, so get pre-tied clip-on or zip-up ties for kids (very easy to find on Ebay in a wide range of colors and sizes) or wear bow ties. You can also try bolo or string ties if that's your style. I like wearing a vertical pin on my shirt placket to give the suggestion of a tie without the formality of one.
My favorite way to dress up a button-up shirt is with a vest; it can be hard to find one that fits if your chest is big, but a little subtle tailoring can turn a blocky square vest into something smooth and glorious that also hides your waistline a bit. Sweater vests stretch to accommodate your shape and are great for autumn. In the winter, wear suit jackets. In the summer, wear a white cotton undershirt (I like Hanes men's small) to soak up sweat and keep your thin cotton shirt from wrinkling or going transparent. Use shirt stays--they're like suspenders that attach your shirt to the top of your socks and keep it tucked in and smooth. A neatly folded pocket square or a pair of suspenders can also do wonders, though I find that I have to wear a binder (gc2b is an excellent brand) if I want suspenders to lie properly on my chest.
And hats! Hats are great! My preferred hat shop is Goorin Bros. even though they inexplicably started dividing their collection into "men's" and "women's"; ignore those artificial distinctions and get yourself a fedora (we can reclaim them from the whiny bros) or a flat cap. If you have long hair, either tuck it up into a hat or wear your hair in whatever long style is associated with men of your race/culture (single ponytail, single long braid, tidy dreads, loose and all one length--there are a surprising number of options even if you don't think of there being long hair styles for men), with or without a hat.
Get shoes from Tomboy Toes, or from Payless, which has some decent men's shoes in very small sizes for very cheap. My first pair of shiny captoes in size 5.5 came from Payless and cost something like $20. I still have them and love them, even though the soles have cracked.
Pay attention to detail. If you're wearing a jacket, shoot your cuffs. Shine your shoes. Wear silk socks and boxer briefs, even if no one but you will know. Iron your handkerchief. Wear earrings that match your cuff links. Wear men's shirts rather than masculine-styled women's shirts so that the buttons are on the correct side, and never ever wear a men's shirt with a women's jacket or vice versa. The little touches that pull an outfit together are really what give the impression of dandyness.