As you can see in the file attached, I am trying to create a simple fidget spinner's shape in Inkscape, I have the basic circles in place but I need to cut parts of the circles where they touch. I have searched for tutorials on youtube but I found intersection and union of objects etc. and haven't been able to put that to use here. Would appreciate any help on this. 

Open objects panel (ctrl+shift + L or object > objects panel).

• In object panel, click while holding ctrl key on path1, path1-0 and path1-0-3 : they become all selected (and highlighted in blue in object panel)
• Do path > union (ctrl + +) : they become one unique path (still named path1 on my PC : this name was the lowest path's name)
• Hold ctrl key and select circle1 (as path1 was still selected, now both path1 and circle 1 are selected)
• Do path > difference (ctrl + -) : on my 1.4 developpement version, the unexpected result is a full black shape on canvas (because fill color is undefined and stroke color is undefined) : don't panic : left shift clic on black color on palette bottom left of screen (this sets stroke color to black) and click on red cross bottom left of screen (to remove any fill).

Edit : the following is rather complicated because in fact your circles don't intersect with the outer shape (the path named circle1).

• I would enable snapping to paths > paths and perpendicular lines and center of objects.
• then I would draw three small paths, actually segments : first point is the center of one bearing, then search for the expected intersection points : when snapping option refers to perpendicular to path, place the other node : repeat for each bearing. let's call these nodes three nearly intersections points
• now enable snapping to cusp nodes and draw a three node path : each node of this path is one of the three "nearly intersection point". Run extension ellipse by nodes (or something alike). You get a circle shape : do path > object to path to cook effect.
• select this circle with circle1 and do path > intersection. let's name the result "the outline"
• now you can see that your bearings are not centered.
• you can fix this : draw a 2 nodes segments around one bearing, where "the outline" shows the "nearly intersection points" : run again ellipse by nodes LPE and cook the effect with ctrl + shift + C. Repeat for the other bearings.
• enable snapping to center of rotations to replace every bearing group on the center of theses three circles.
• select these three circles and "the outline" and do path > union.

I said it was a complicated way...

As noted above, this would be easy to use the shape-builder, if the circles actually touched.

I´d start with a triangle and a rectangle and 2 Path Effects I guess:

I spent some time to make something less complicated, but Polygon's method looks even more straightforward.

Here is mine.

Simpler, maybe.

Thank you for all the help. I tried @David248's second method (the first one was complicated for me) and it worked great. Still working on the colors. You all are really good at this. Thanks again for the fast replies. 😄