This is an example logo. I have the original AI file, but I want to update it such that it has has rounded edges. I also want to round the look of the arrow, but I don't know how to do any of this.
Rounded edges can be presented only with shading. You need more colors.
You can apply in Illustrator Effect > 3D > Extrude & Bevel with a round bevel.
I have only a screenshot. Thickness and rounding was easily added to it in Photoshop as layer style Bevel and Emboss. Before adding it the white background was erased and the black color was adjusted lighter to make darkening possible:
ADD : Actually the main color of the text can still be black, if you have light from two directions. In Photoshop's Bevel and Emboss you can use white screening for higlights and shadows:
The blue arrow can still have normal shadows. It can have the effect separately with different light and other settings: