Learn Russian: У меня есть три идеи.

Because of the number 3 (три). In Russian, when you count 2, 3, or 4 of something, the noun that follows must be in the genitive singular form. The dictionary form is идея, and its genitive singular form is идеи.

Doesn't the -и ending on идеи mean it is a plural form?

It looks identical to the plural! For feminine nouns ending in -я like идея, both the plural and the genitive singular forms happen to be идеи. However, the rule for the numbers 2, 3, and 4 strictly requires the genitive singular, so grammatically, that is the specific form being used here.

I learned that "I" is я. Why do we use У меня to say "I have"?

Russian usually expresses possession by saying "at someone there is" rather than using a verb for "to have." The word У means "at" or "by," and it forces the following word into the genitive case. This changes the pronoun я to меня.

Since we are talking about three ideas, why isn't the verb есть plural?

In this possession structure, the word есть translates roughly to "there exists." In modern Russian, есть does not change form for plural objects. You use the exact same word whether you have one idea or three ideas.