I recently transited Houston and can share my experience.

On your outbound leg after arriving on Delta, you need to collect your bags and then look for inter terminal train to go to your departure terminal. On your return leg, as per US law bags need to be collected on arrival, and you need to recheck them in on United. Since you are holding two separate tickets, the bag handover will have to be done at UA check-in desk in transfer area. Kindly ensure sufficient connection time in Houston (Minimum 3HRS is a must). During my travel, UA check-in was painfully slow, as they had very agents supporting a huge crowd. Additionally, it can take anywhere between 30MIN to an HR to clear security during peak hours.