Learn Hardcore Italian: Trovo un'ombra fresca in giardino.

Because it comes from the feminine article una. When una is followed by a word starting with a vowel, the final a is dropped and replaced by an apostrophe (un’). So un’ombra is the elided form of una ombra.

Why do we say Trovo instead of another verb like Vedo or Cerco?

Trovo means I find, expressing that you come across something (in this case, a fresh, cool spot). Vedo (I see) would merely describe noticing it, and Cerco (I look for) would focus on the action of searching rather than finding. Trovo indicates both that you are looking for shade and that you successfully come upon it.

Is it possible to say nel giardino instead of in giardino?

In Italian, in giardino is a perfectly natural way to say in the garden. Using nel giardino (literally in the garden as well) is also correct, but it can sound slightly more specific, referring to a particular garden. In giardino is more colloquial and commonly used in everyday speech.

Does ombra only mean shade, or can it have other meanings?

While ombra primarily means shade or shadow, it can have other figurative meanings in Italian. For example, it can refer to a small amount of wine (in some regional uses) or be used metaphorically (like an ombra di dubbio, a hint of doubt). In most contexts related to sunlight, it just means a place sheltered from light or heat.

What tense is trovo, and how is it conjugated?

Trovo is the present indicative, first-person singular form of the verb trovare (to find). The present indicative conjugation for trovare is:
• (io) trovo
• (tu) trovi
• (lui/lei) trova
• (noi) troviamo
• (voi) trovate
• (loro) trovano