For your example, you'd say
I will neither buy groceries nor eat at a restaurant.
In the general case, the rule-of-thumb is to phrase the sentence as "I will neither X nor Y", where X and Y are phrases containing verbs that could independently form the two sentences "I will not X" and "I will not Y". (In our sentence, X = "buy groceries", Y = "eat at a restaurant").
In case both X and Y begin with the same verb (say buy), you can pull out the verb so that it distributes over the neither-nor construction:
"I will neither buy groceries nor buy shoes." —>
"I will buy neither groceries nor shoes."
"I will neither eat in the park nor eat at a restaurant." —>
"I will eat neither in the park nor at a restaurant."
"I will neither do this nor do that." —>
"I will do neither this nor that."