When I travel, I do like to see many of the popular sights, but I also like to find some out of the way places. One way to engage in cultural immersion is to take yourself out of the very tourist-heavy areas. You can wander along the streets in other areas of cities or you can choose to visit small towns or villages in between the larger ones. This is one advantage of traveling by yourself and not with a large tour group.
Another suggestion I have for cultural immersion is to really open up to cultural experiences. Eat, drink and be merry like the locals. When in Rome, right? You may not be keen on trying some of the foods or customs, but you also may never have the chance to again. You may just gain a better understanding of the people while you're at it.
Lastly, if you are very brave, you can immerse yourself linguistically. Try not to speak English along the way! It's hard, especially when people volunteer to speak English with you, but will pay off in the end.