Entries field: Output entries related to category
The targetElement and sourceElement take either an ID, an Element Model, or an array of either.

So, in your example, you'd want to do:

{# Categories #}
{% set CatToDo = craft.categories.slug('things-to-do').first %}
{% set CatToEat = craft.categories.slug('things-to-eat').first %}

{% set toEat = craft.entries.section('yourSection').relatedTo({
   targetElement: CatToEat,
   field: 'activityPicker'
}) %}

[...]


Notice the .first in there as the difference. You will only want the one category since you're passing the slug.

Note: This assumes that CatToEat is indeed the targetElement. See here for more information: http://buildwithcraft.com/docs/relations#the-relatedTo-param