So a straight bond is a bond without any options. A bond without any options is invulnerable to interest rate volatility.
Higher volatility = higher call price = lower callable bond price as investors are short the call option. On putable bonds, the effect is the opposite as investors are long the put option. So if volatility goes up both options will go up in value but the callable bond will fall and putable bond will rise in value. Rising interest rates will lower the call price as the issuer is less likely to exercise it due to lower bond prices. The effect on put options is the opposite as higher interest rate = higher uncertainty of cash flow = more likely to be exercised by the investor.