I think it depends a lot on the project, but usually I try to find a relevant starting point, maybe even just a single colour, and then build on that foundation with tints, shades, adding complementary or analogous accent colours, and so on.

For a creative/promotional/marketing kind of web site, there are often prominent images, and those can suggest a colour palette. Advertising a get-away-from-it-all-break to a tropical island? You’ve got sky blue, sea green, tree green, sandy yellow, and so on. Sample a few representative colours from the photos and you’ve probably got a good starting point.

For more business/professional sites, if the brand has a logo that already incorporates colour, again following its lead is often a good beginning. Even if it only uses one colour, chances are that colour or some variation of it can be used as your first accent colour, or part of the background, or even the main colour for body text. If there are multiple colours in a client’s logo, it would take a solid argument for me not to follow the existing colour scheme in other contexts.

If there are no relevant images or existing branding guidelines to suggest any colours you can start with, you need to be more creative. You can start with what your client does or makes, but personally I prefer to concentrate on the kind of impression they want to make rather than the market or industry they’re in.

For example, suppose it’s a technology firm. For the trustworthy, businesslike look of a professional consultancy, it’s hard to go wrong with a cool and darkish palette, probably built around blues and perhaps with elements of green or purple. For a firm that makes custom keyboards and controllers for serious gamers? Almost anything but that kind of businesslike look, but probably something dark and edgy, maybe black with silver accents, plus a bright accent colour like a red, purple, or electric blue/turquoise. If the firm makes heat recovery and ventilation systems? That might suggest hot and cold, so maybe a palette with some red/orange/yellow colours plus some light blue/cloudy grey, probably in those combinations, and probably with specific colours that lend themselves to warm and cold gradients or being set as contrasting/complementary pairings in the visuals.

That’s all thinking about promotional/brochure kinds of material. For some contexts, such as colours for data visualisations or user interfaces, more practical concerns tend to dominate. Good maps are often a source of inspiration for me if I need one of these practical schemes, as they tend to emphasize things like a range of colours that are clearly distinct but not clashing, or twin-hued palettes with a lot of tints and shades to show data that varies on a scale with some significant threshold. I’d probably still start by looking for one or two relevant colours to anchor the scheme, though, say a sea blue vs. a grassy green or mountain red/brown for a map visualisation showing height above or below a waterline or similar threshold. But beyond that, for these practical cases, I tend to be a lot more systematic about generating further colours to fill out the palette, for example interpolating between two colours in some perception-based colour space and choosing as many other colours as I need spaced around evenly along the line.