Although I've made plenty of Genovese (green) pesto before, I've only tried out red pesto once that I can recall - the first time I cooked for the Eldorado girlies.
It went down pretty well I seem to remember, along with the multi-coloured home-made pasta that they insisted on making.

It's instructive to make your green pesto the traditional way by using a pestle and mortar or chopping everything finely with a knife or mezzaluna at least once, but forever after you can just blend everything in a food processor like a normal person.

Incidentally - bear in mind that Parmesan (and Grana Padano and Pecorino) are not vegetarian, so if you want veggie pesto you'll have to find a substitute hard cheese that doesn't use calf rennet to curdle the milk.

Ah, Genoese Pesto - there must be as many versions as there are Italians, so don't feel too constrained by the quantities.
Or the ingredients.
You essentially need basil, garlic, pine nuts, Parmesan and olive oil (and not all purists would agree on those basics).

Traditionally the garlic and pine nuts would be ground in a marble mortar with a wooden pestle, followed by the basil and coarse salt and ground until creamy. Then the cheese and olive oil added.
You can also chop the ingredients together using a knife or mezzaluna, but to be honest these days it's easiest just to use a food processor on pulse. Be careful not to overblend though.

You can argue about the cheese if you want, whether to use Parmigiano-Reggiano (Parmesan), Grana Padano, Pecorino (Romano or Sardo), or some combination of them.
You can try substituting other nuts like cashews, walnuts, almonds or even peanuts, you can toast the nuts first too if you want.
You might roast the garlic or leave it out entirely.
You can substitute rocket (or other herbs) for some or all of your basil if you're American.
You might, like Classic Italian cookery writer Marcella Hazan, beat in a couple of tablespoons of softened butter at the end.
Or you might want to add a few non-standard elements like mint leaves, lemon juice, even anchovies or a blob of yoghurt (apparently!).

Personally I like about 100g basil, a half-dozen mint leaves, 50g half pecorino-half parmesan, 1 garlic clove, 2 tablespoons of toasted pine nuts, about 200ml of extra virgin olive oil and possibly a splash of lemon juice if I'm in the mood.

Fills a jam jar

Toast your nuts (if you want) in a dry frying pan or in the oven until they just begin to darken.
Roast the garlic (if you like, and can be bothered).
Pick off the basil leaves (only the small and tenderest - ideally), and wash and dry them thoroughly.
Grind up your garlic, nuts and then the basil leaves (and a few mint leaves) using your preferred method. Having done it once with a knife I now just pulse them up in a food processor using olive oil for lubrication, crushing the garlic and roughly chopping the basil first to help things along, but you can grind them in a pestle and mortar with a bit of rough salt if you're a masochist.
Mix in the grated cheese and enough olive oil to make a thick paste.
Add a splash of lemon juice, a spoonful of yoghurt or grind in some anchovies if you like.
Season to taste.

Pour it into a jar and cover with a little extra oil. This should keep for months in the fridge, if you can resist its siren call.
Yep, it's lovely.

Pesto Rosso (Red Pesto)
Sun-dried tomato and toasted walnut pesto
Sicilian red pesto 'Pesto alla siciliana' is similar to pesto genovese but substitutes almonds for pine nuts, adds tomato and uses a lot less basil.
Sophie's version here is another take on the sauce.

Toast the walnuts:
Either toast the walnuts in a large frying pan, or spread them on a baking sheet and bake in a 200°C/Gas 6 oven for 3-4 minutes. Shake them once or twice until nicely browned, but don't let them burn.
Shake the toasted walnuts in a wire sieve over a bin to remove their loose papery skin, then leave them to cool.

Place the nuts, sun-dried tomatoes, garlic, Parmesan and chilli, if using, in a food processor and process until smooth, gradually trickling in enough oil to give a creamy sauce. You can use both olive oil and some of the sun-dried tomato oil here.

This should keep for a couple of weeks covered with a layer of olive oil in a jar in the fridge.
Feel free to add basil, parsley, celery, lemon rind or even olives to the mix if you like.

Rocket and Parsley Pesto
I made this (with a bit less cheese) to serve with osso buco instead of gremolata, and to be honest also to use up some leftover rocket.
But it makes a pretty good pesto too - though I didn't add any nuts this time, I reckon walnuts would work well, if you were so inclined.

Makes about 150ml

Grate the peel of a lemon and the Parmesan and set aside.
Pick the parsley leaves, wash and dry thoroughly. Wash and dry the rocket thoroughly.
Blend the leaves with the olive oil, garlic, and juice of a lemon.
Mix with the grated peel and cheese.
Zesty mixture.