I am working on creating a website logo and also t-shirt designs for my clothing brand to sell on my website using Adobe illustrator. I am allowed to use the fonts already built into illustrator for both those uses? If not, what are my options?
There aren't any fonts built into Illustrator... this is a common misunderstanding, and really confuses people when they change computer. Illustrator presents the fonts you have purchased and installed on your computer, and those that came with the computer. Same fonts you'd see in Word or whatever. PLUS, Creative Cloud subscribers can load fonts from the Adobe Fonts service.
Each font has its own licensing terms and contract.