I'm making a logo and I want there to be curved text on the inside of the oval. How can I do this? (Please excuse me as I have barely ever used Inkscape but I am trying to learn.) Thank you!

Assuming you want the text running along the inside top edge of the oval (ellipse, in Inkscape terms), and that you want it to remain upright, here's what I would do:

1) Duplicate the ellipse and give it a different stroke colour for clarity
2) Resize the new ellipse, holding Shift to resize from the centre. Adjust it so that the top curve of the ellipse follows the baseline you want your text to take.
3) Double-click on the ellipse to access the editing handles, then use the circular ones to reduce your ellipse down to an arc that will be the baseline of your text.
4) Select both that arc and the text, then use Text > Put on Path.
5) You can still edit the start and end handles of your arc to position the text exactly where you want it.
6) When you're finished, remove the stroke from the arc so that it's not rendered. (If you need to edit it again, use View > Display Mode > Outline to get hold of it, then give it a stroke again)

Sorry I didn't specify where I wanted the text. I would like it on the bottom inside of the elipse. Is is basically the same process?

Essentially. But you'll probably find that the text appears on the outside of your baseline path. If that's the case, change the arc back to a full ellipse (there's a button on the toolbar for this), flip the ellipse horizontally (press the H key), then continue from step 3.