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A bit off topic here but I do hope you get time to try out your new camera before embarking on a major holiday.
I personally use the Lumix in P mode. Because the lens is a bit on the slow side there isn't much sense of using manual as you don't have a huge aperture range to work with. Although you can shoot in RAW I just use the JPEG Fine setting which is good for most situations.
Lumix has an excellent Macro setting which you can use to get those flower photos in the greenhouse. I avoid flash except to fill in the case of backlit subjects. I don't use Face Detection unless I want to track a person in video - not very often.
The normal AF is fast and quite reliable. My ZS50 sometimes takes its time focusing in low light such as inside a church. You will probably do better with the TZ220 since you have a larger sensor.