How much sugar and vanilla extract should I use in 1/2 gallon of ice cream?

You're making what is (at least to Americans) called Philadelphia-style ice cream. The name makes it fairly easy to find the ratios. Checking three recipes:

• http://www.culinate.com/books/collections/all_books/The+Perfect+Scoop/Vanilla+Ice+Cream
• http://www.seriouseats.com/recipes/2014/12/30-minute-philadelphia-style-ice-cream-recipe.html
• http://www.williams-sonoma.com/recipe/philadelphia-style-vanilla-ice-cream.html

all agree on ¾ sugar for 3 cups of dairy. So you'll likely want thrice that, or 2¼ c. sugar. (Sugar in ice cream isn't just for taste, it's also very important for texture.)

Then you'll want somewhere between 1–3 tablespoons of vanilla extract. This is a wide range, but actually vanilla ice creams available commercially in the US vary widely in the amount of vanilla flavor in them. I would recommend, if you remember what the melted ice cream tastes like (not frozen, as freezing will lessen the flavor somewhat), that you start with 1T, taste it, then mix in more as required.

Consider also a little bit of salt. Also, a 2:1 recipe of heavy cream to whole milk is common—saving you from having to purchase half and half.

I was already mostly done with my research when the other answers posted (thanks!) and they suggest quantities that match mine, but I don't want to lose the extra information I gained, so here's the results of a survey of ice cream recipes.

I looked at the first page of results on Google searching for "vanilla ice cream" and recorded 3 non-custard recipes, and 6 custard recipes. I recorded the liquids, sugar, extract, salt, and egg yolks (in the case of custard) and found the following:

Sugar

The range was fairly large, going from 1 cup sugar per 6 cups liquid on the low end to 1 cup sugar per 3 cups liquid on the high end. The average was right around 1 cup sugar to 4 1/4 cups liquid.

Vanilla

This was a little more variable. On the low end some recipes called for 1 teaspoon vanilla extract per 4 cups liquid, and on the high end some called for 1 teaspoon per cup of liquid. The average here was 1 teaspoon per 2 cups liquid.

Interestingly, unlike sugar, custards and non-custards differed a little in the amount of vanilla used, with custards using less. Non-custards ranged from 1 teaspoon per 1-4 cups of liquid, averaging to 1 teaspoon per 1.5 cups liquid. Custards ranged from 1 teaspoon vanilla extract for every 1 1/2 to 3 cups, averaging to 1 teaspoon per 2 1/4 cups.

Salt

When it was included, it ranged from 1/8 teaspoon per cup liquid on the high end to 1/8 teaspoon per 13 cups liquid. The average is 1/8 teaspoon per 3 1/4 cups liquid but 1/3 of the recipes didn't include it at all.

Egg Yolks

For the custards the egg yolk recommendations varied widely, from two egg yolks per cup of liquid down to one egg yolk per 3 cups liquid. It looks like there's a lot of opportunity for one to choose a ratio that suits their taste and texture desires. The average was one egg per 3/4 cup liquid.

For 9 cups of liquid (about 1/2 gallon prior to churning)

• Sugar: between 1 1/2 cups and 2 3/4 cups. Average is 2 1/8 cups.
• Vanilla: between 2 1/4 teaspoons and 3 tablespoons. Average is 1 1/2 tablespoons (4 3/4 teaspoons)
• Salt: if included, between a pinch (1/16 teaspoon) and 1 teaspoon. Average is 1/4 teaspoon.
• Egg Yolks (custard): between 3 and 18 eggs. 11 is average.

A typical vanilla ice cream recipe looks like this:

• 2 cups Heavy Whipping Cream
• 1 cup Whole Milk
• 2 Large Eggs
• 3/4 cup Sugar
• 1 teaspoon Vanilla Extract

So roughly, that's 3/4 cup sugar and 1 teaspoon of vanilla per 3 cups of liquid. Multiplying that by three for your recipe, and you get 2 1/4 cups sugar and 3 teaspoons (1 tablespoon) vanilla.

It should be at least close enough that you can taste it and make any adjustments necessary before churning.

Just a simple answer without all the formulas.

Most ice cream for a half gallon will have 3/4 to 1 cup of sugar and 1 teaspoon of vanilla extract.