If your whole story is a first-person narrative, no, no quotation marks for that. Use quotation marks for actual speech of your focal character and others.
Different novelists use different devices. For actual thoughts, some use italics, but this device can get very tiresome, especially if there is a lot of it. And there may sometimes be a fine line between the character’s narration and an actual thought, since it is all, in a sense, his or her thoughts. Don’t start something that’s only going to get in your way and annoy the reader.
Some writers start a new paragraph for internal monologue. Some express actual thoughts without quotes, but set off, like this:
The rain was coming down heavier now. Too wet to go out, I thought.
If your character speaks to himself out loud (“Oh, no!” I cried.), yes, use quotation marks.