For moderation, go to Linear Regression panel in SPSS. Enter your dependent variable first. Then, select all your independent variables excluding the moderator. This is Block 1 of 1. Click Next on up-right and now enter the moderator along with the other independent variables. Leave the Method as is (Enter); under the Statistics tab, select R squared change in addition to what have already been selected (Estimates and Model Fit). Now, you are ready to go. Note that you may need to center one or more of the variables in hand in further analysis. Follow this path for that: Transform --> Compute Variable.
For mediation, on the Linear Regression panel in SPSS, select your mediator as the Dependent and your IV as your Independent(s) and run the regression. Then, run another regression with your actual Dependent Variable and your IV. In the final step, regress your actual Dependent Variable both on your mediator and the IV. This is the procedure proposed by Baron & Kenny (1986).
Andrew F. Hayes' textbook "Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach" is an extremely helpful resource. He created a macro in SPSS called PROCESS that performs mediation and moderation. You can download it for free from here: http://www.processmacro.org/download.html
It may be helpful to dig around online to understand how to use PROCESS, however I do recommend buying his book as he gives great examples of mediation, moderation, mediated-moderation, etc. (it's relatively cheap- $39 on Amazon).