As a general rule, wheel nuts are 14mm front, 15mm back. 8mm or possibly 10mm for adjusting brakes. Seat post clamp, if it's got a nut rather than an Allen key head, will likely be 13mm but could be 12mm or 14mm. 17mm should fit the locknuts on the back axle, slim cone spanners for the bearing cones usually fit a variety of sizes. 1 1/4" AF seems to fit most threaded headset locknuts but a large adjustable spanner will do. Pedals will typically require a 15mm open end spanner and dedicated pedal spanners are available.
If you're working on a vintage English 3-Speed, throw out the rule book as everything is most probably BSF/BSW.
Assuming you're working on a reasonably modern bike, I would just buy a set of metric combination spanners. Sets typically cover the range 6mm to 24mm. Some you won't need for bike work but it's still probably cheaper than buying the ones you do need individually. Buy the best you can afford. A set of spanners can be a lifetime purchase, poorly fitting cheap ones can skin your knuckles and ruin the nut.