How many tradelines do you have?
Removing bad stuff is the most powerful thing you can do!!!
Good idea to start building. The credit cards will help you in the long run, 2 better than 1. I am a firm believer that even 4 are better then 2. New credit drops scores in the short term for many months. In the long term the credit cards will impact your score nicely.
To reach your goal in the short term you will need to remove the bad things. Remove enough and very good scores are possible. Realistically you can do great improvements over 5 months. It will feel at times, nothing is happening. Keep at it, things do work out.