In the early credit cycle bonds & stocks both go higher since lower interest rates mean easy borrowing for companies & of course lower rates mean higher bond prices.
In the late cycle (which we seem to be in) bonds definitely go lower but stocks are a mixed bag. Usually in the early days of the rate rise from the Fed stocks go down but if the rates don't create a lot of slowing of the economy then stocks go up but the sector make of the stocks changes.
Lower rates means tech, retail, REITs. etc. should do well but in a rising rate environment Energy & Banking do well along with stodgy companies that don't require external funding.
edit: I added the following
When the recession comes bonds traditionally rise since it is seen as a safe haven -- this is really only true for Sovereign Debt and not corporates. I believe however this time I'm not sure a downturn would create a lower interest rate so I have a hunch that the US dollar is actually the new Soveriegn Bond.