Proper usage of pronouns

As an ESL student, sometimes I wonder whether I use too many pronouns. For example:

She grabbed her purse, she took some money and then she went to buy some groceries. After that she returned happily with a lot of things she bought.

Can it be changed to something like this?

She grabbed her purse, took some money and then went to buy some groceries. After that returned happily with a lot of things she bought.

When can you get rid of the pronouns without losing the sense of the person or object you’re referring to? Where can I get more information about this?

You need at least a subject and verb in each sentence and independent clause. Sometimes the independent clauses in the same sentence can share a subject. As you suggested

She grabbed her purse, took some money and then went to buy some groceries.

There are three independent clauses in this compound sentence.

She grabbed her purse,
[she] took some money and
[she] then went to buy some groceries.

The second and third clauses are all sharing the subject she. The use of fewer pronouns here makes the sentence flow well and sounds less formal.

The second sentence is a problem. In your example, there is no subject.

After that [subject?] returned happily with a lot of things she bought.

The dependent clause she bought modifies a lot of things. It cannot share its subject with the main clause. Instead, it should read

After that she returned happily with a lot of things she bought.

The second pronoun should not be omitted either. Dependent clauses generally need their own subject.

You can safely remove pronouns if one noun or pronoun governs multiple clauses, as in your first sentence:

She grabbed her purse, took some money and then went to buy some groceries.

You would do this to write in an economical style, which is easier to read. However, if you want to emphasize each separate clause as a distinct idea, you might want to leave the pronouns in:

She grabbed her purse, she took some money, and then she went to buy some groceries.

This has a more dramatic feel to it.

The simple rule is to leave out a pronoun when the antecedent is clear, and to leave it in if not. You have to be good enough at English to tell when something is clear and when it is not.