Oh for goodness sakes... you know what to do with a carrot.

Peel it, slice it lengthways into quarters, cut into two inch pieces and then steam it for as long as you feel. Start off by doing it for, let's say, 4 minutes, by which time it should be quite soft. Leave to cool or run under the tap, etc etc then give to the bub.
 
Apart from a peach which my baby grabbed from my hands, steamed carrot was her first food. To begin with she didn't have the knack of chewing or gumming her food so she would lean forward and bite a big chunk off which would promptly fall straight out of her mouth. It was only when I saw some tell-tale orangey 'material' in her nappy that I realised anything had gone in at all.

Carrots are great to take out with you as they aren't terribly messy, and as the babies' teeth start to come in they will enjoy chomping on some raw carrot straight from the fridge. Not a lot of the raw stuff goes in, so it's really for the teething.

Post Script October 2006
Actually, what I've discovered is that carrots are brilliant steamed for starting off, then great raw when those very first teeth start coming in. However, once Babybear began to get the hang of biting (by which I suppose I mean hanging onto the carrot and wrenching it hard out of her mouth) I began to get a bit worried about choking hazards again. So if she gets raw carrot I now give it to her in big fat discs. Although she's gone right off her former favourite so it's a moot point at the moment.

Post Post Script March 2007
And now she's back on the raw carrot again. I suppose it could be considered a choking hazard still, like everything else. But I wonder if the fact that she's eating it as if it's a sweetie outweighs the potential for a wee cough? Anyway, she never has so I'm not thinking about it just now. She likes the lengths quartered and just bites little bits off with great delight. Oh, and she also thinks it is delicious in spag bol, chilli, stir fries etc. It's weird how they go on and off things...

Top and tail them then steam them over boiling water for about 4-5 minutes. Leave to cool or run under cold water. Try one yourself to check that they're not too hot inside then give one to your baby. So cute, what normally happens with my bub is she chews the outside until it goes kinda stringy and sooks the beans out of the inside. Too gorgeous. And hardly any mess, hurrah!

Cut your broccoli up to make 'trees', but not too small. It's best, I have found, to leave as much of the stalk as possible to be used as a handle, but if you think that it makes the 'tree' part too big then slice up through the stalk to half or even quarter it lengthways.

Steam your broccoli over some boiling water for 6-7 minutes then leave to cool (or run under cold water if you are in a hurry, but remember that if you feel it's too hard still that this will stop the cooking process).

Give it to your child and if they are good and clever they will hold it by the 'handle' and eat the 'tree'. My child resolutely refuses to do anything but dig her fingers into the branches of the tree part and chew on the stalk, which causes the top part to disintegrate in the most alarming fashion and to fly across the room. It goes EVERYWHERE. You know those chubby little folds of fat that you love to tickle? In there. And that beautiful, expensive high chair* you bought? In every seam of the the fabric seat, every hinge, every corner. Broccoli, my friends, is very much a pre-bathtime vegetable. But delicious and healthy and oh-so enjoyable for the babies.

*Please see Top Tips on buying a highchair.

Wash, chop into pieces about two inches long and then slice the pieces into wedges. Remove the seeds if you like. I don't tend to, it depends how fresh and young you think the courgette is (and how fresh and young you are feeling at that particular moment).
Really handy to take out with you as they don't make much mess, not a great deal gets eaten but the babies enjoy sucking and chewing on the sticks and it seems to help their teeth.
I have also given my baby some steamed courgette, which she enjoyed and a lot more went in, although if I'm honest she likes it best lightly fried in butter. Just like her parents...