Non-destructive Edit History
The bad news is that the non-destructive edit history is not something that can easily be moved between Picasa and any other photo processor. Sorry, you just won't be able to keep the history of edits nor will you be able to necessarily recreate the identical effect in any other software without painstaking comparison and fine tuning.
Originals and Edited Images
The good news is that Google Picasa does save the original unedited images. You can simply right click on an edited image and select Locate> Original on Disk
to track down where Picasa has decided to store your originals, although the pain point is that this is not one centralized location - they are stored in a subfolder of the original location.

One you have located the originals, you could import both the originals and the edited images to Lightroom so at least you didn't lose what you started with nor the final output. Note that to utilize some of these features, you do have to Save
the image in Picasa (the original isn't moved to the secondary location until you save).
Facial Tagging
Facial tagging is a special area because as I understand it there is not standardized way to store this information in the Exif or IPTC metadata, or if there is - it isn't being used widely enough to be valuable in this instance.
The good news is someone has done the hard work and created a plugin to save your facial tags. Jeffrey Friedl's Blog - Jeffrey’s “Picasa Face-Recognition Import” Lightroom Plugin.
Other Areas
There are other concerns that most will have moving between Picasa and other options. Picasa stores a few things in it's picasa.ini files, such as the star rating and facial recognition information. You could come up with a programmatic way to parse out certain pieces of each picasa.ini file, or you can try a bit more manual but still reasonable approach of extracting the information into keyword metadata. For example, if you want to save your star rated photos and tag them as such for later application in Lightroom, you can follow something similar to this:
- Filter your images in Picasa to only show the Starred images:

- Select all images and add an appropriate keyword to all of the images:

- Import images to Lightroom and you will find the new keyword. This can now be used to drive collections, specific edits, addition of star ratings, etc:

This same idea of adding keywords could be extended to a wide range of uses beyond just star ratings although it likely would have to be an even more manual process. Again, if you have the know-how and time you likely can pull some of this data out on your own with scripting, and others have attempted this and had some success(example). So you may want to browse for specific solutions if you have a need and the manual solutions would be too time consuming.
Final Thoughts
The good news is that frankly Picasa can still live on for some time. It isn't going to self destruct or uninstall itself. Picasa 3 will work on your computer for as long as it remains compatible with your operating system (OS), and the newest build is compatible with Windows 10. So take some time to get comfortable with new tools such as Adobe Lightroom, and get all of your necessary data moved over - but don't worry about doing it yesterday. Who knows, it is possible in a short time that even more tools will be developed to ease the transition.