Once you use a straight edge to keep the cuts square (or as square as possible), you also need to really be critical of your measurements. Make sure to account for the saw blade and measure from the blade edge to the straight edge.
When making multiple cuts with the same dimension, I usually take the first piece I cut (after I have made sure it is the exact dimension I need) and I draw a line on the wood I'm going to cut by putting the first cut piece on top. I then check that measurement and make sure it's correct and then set up my straight edge to make sure that the blade sits correctly on the outside of the line. Once you're set up, you just rip as steady as possible and make sure all sides are supported well so you don't pinch the blade.
Without a table saw, I have found that method to work the best for me but it is very time consuming. What takes 20 minutes on a table saw, takes me about 2 hours with a circular saw.