What you have is a 2 x 2 ANOVA with one within-subject factor and one between-subject factor. It would not be appropriate to test the same dependent variable with an ANOVA and a Chi-square test. ANOVAs are for ratio variables and Chi square is for frequencies.
To conduct your ANOVA in SPSS you want to start out with three columns of data. One column with all of your pretest scores, one column with all of your posttest scores and one column indicating whether the scores in that row are from the experiment or the control group (you can just use 1 and 2).
To perform the analysis go to the repeated measures ANOVA. First it will ask you to indicate your repeated measures variable. Create a label (like time) and give it 2 levels. Next you want to fill in those two levels with your pre and post test scores. Put the remaining condition label (experiment vs control) in the "between-subject factor" box.
The output will give you:
1. A main effect of test time: Is the difference between pre and post test scores significant?
2. A main effect of condition: Are the experimental group's scores different then the control group's scores?
3. An interaction: Is the difference between the pre and post test scores the same for the experimental group as it is for the control group?