Transportation to and from Chinatown can vary depending on your prefered method of choice. If you live close enough, you could definitely walk, but otherwise the CTA Red Line, driving by car, New York Central Railroad and the Dan Ryan Expressway could all bring you there, if not at least very close to the boundary of Chinatown.
Both Chinatowns, new and old, have a variety of restaurants and shops you can visit for food and drink, such as Joy Yee, a Chinese resturant chain that also sells things like frozen yogurt and bubble teas. Or Ken Kee, a restaurant serving Hong Kong street food as well as Cantonese cuisine, everywhere you look in Chinatown, there are restaurants selling food and drinks everywhere, it just depends what you want that day.
Besides the obvious (shopping at Chinatown's many gift shops), there are many activities to do there such as visiting the Chicago Chinese Cultural Institute, offering tours of Chinatown and founded so that it could bring Chinese language and cultural programs to libraries and schools, and similar to that is the Chinese American Museum of Chicago, allowing visitors to learn more about Chinese culture and explores the journey immigrants take traveling to the Midwest- including stories personally submitted by those living in Chinatown, and the Heritage Museum of Asian Art, showcasing different Asian artforms with pieces of art that date back hundreds (if not more) years.