This is easily doable on your own since transport infrastructure in most of these places is better than in the USA and user-friendly. If I had to dispense with something on your wish list for this trip, I would get rid of Guilin due to geography and the fact that April isn't the greatest weather month for there (rainy). With that out of the way, Beijing + Xi'an plus Shanghai and any two of the other cities you mentioned (Nanjing, Suzhou, Hangzhou) become doable. I think you should keep Beijing in the mix, and fly into either Beijing or Shanghai then out the other. Sample in the southbound direction (works the other way just in reverse):

Day 1 Arrive in Beijing
Days 2-4 Beijing
Day 5 morning fly or bullet train to Xi'an; aft Xi'an
Day 6 Xi'an
Day 7 morning fly to Nanjing; aft Nanjing
Day 8 Nanjing and late aft fast train to Suzhou (only takes about 1.5 hours)
Day 9 Suzhou
Day 10 midday fast train to Shanghai (about 30-45 minutes) or car/driver.
Day 11 Shanghai
Day 12 Shanghai and departure

With an extra day or so, adding Hangzhou becomes manageable. Or you can substitute it for either Nanjing or Suzhou. None of the three cities are the same so prioritizing in the face of limited time becomes a matter of your interests. Nanjing especially for 20th century history; Suzhou for classical gardens; Hangzhou for West Lake and a pleasant environment though not as many high-value "sights" for the foreign visitor. You should also research Pingyao and Luoyang since they could be on your route....but adding them to what you have already means another 3 days.

As for escorted tours, they do come with the downsides of having to subordinate your preferences to the pre-set schedule, poorer meals (usually), and risk of getting stuck with annoying guides and tour mates. So it's up to you as to whether the value is there for you. On this forum we tend to encourage more independent thinking and doing, also doing the intercity travel oneself then using local driver/guide on an as-needed basis per stop, customized to what you want to see/do--yes this does mean that you at least need to crack a guidebook for ideas or research on the internet but that's half the fun. As for intercity travel--if you can get yourself between cities in the USA or Europe, you can manage to do it in China, too. If you commit to doing it without an escorted group tour, focus on the overall itinerary then we can help you flesh out the exact transport connections, the daily play-by-play of sights and how to get around or get assistance getting around, accommodation recommendations, etc.