What is the best way to interpret changes in Treasury yields?

What does yield actually represent?

Yield for a bond is the effective interest rate you get from the cash flows (coupons + final payment) of the bond. It is meant to be mathematically equivalent to putting the money you'd pay for the bond in a savings account that pays the "yield" in interest, and depositing all of the coupons in that account, with the final account balance equal to the face value of the bond.

Do traders who trade bonds trade the yield or the face value?

(I assume by "face value" you mean price since the coupon is fixed). Bonds are compared by their yield, but actual trades happen at specific prices. Traders may quote bonds based on their yield, but at the end of the day they pay/get a specific price for bonds that they trade.

Does the yield go up because bond prices go down or do bond prices go down because yields go up

Trades occur at a specific price, and that determines the yield.

Yields are typically quoted as absolute changes, not relative changes. So if you hear "10 Year Treasuries are up 5 basis points", that means that the market is paying less for these bonds, and the yields are actually 0.05% higher (e.g. from 2% to 2.05%), not a relative increase like you typically see for equities and other instruments.

Note that there's not an exact equivalent of "yield" for equities because they aren't designed that way. Some equities pay dividends but dividends reduce the price of the equity (which is another discussion) unlike bonds. Equities are traded based on an expectation of their actual value increasing, not strictly because they pay a dividend. (preferred equities are equities that act like bonds, but that is also another story)

To answer the TL:DR, you could say that the yield is 10% higher today than it was yesterday.

Traders often use basis points to express changes in yield, to avoid the confusion in terminology that you are experiencing. This textbook chapter [PDF] may help you.