First thing I will say is 'WELL DONE' in the first place for having a go at such a shot. Many people shy away from a shot like this because of the management of all of those people but apart from one or two small children, everyone seems to be looking at the camera. The first ctiticism is that the scene is a bit cluttered and before you say anything about not having much room, you could have removed (or used) the two chairs in the foreground and removing the three white napkins from the table would also have helped, white is always a distracting element in a shot when it isn't the main subject.
The image has a colour balance issue because the faces of the people are very much the same colour as the surrounding wood and the white things aren't white, they are yellow. You can set a custom white balance before you take the shot if you are using Jpeg but if you want to adjust white balance correctly in post production you need to shoot RAW rather than Jpeg. not sure about the Panasonic Lumix but i would think that shooting RAW would be the best option because I'm not sure if cistom white balance will be available or not. The other issue is the depth of field, your shot looks reasonably sharp but if you want to shoot groups of people three or four deep like you have here then you need a greater depth of field otherwise the people in the front row (or wherever you focused) will be sharp and the others will start to go out of focus the further away from the focus point they get. I can assure you that if you'd used an DSLR with a 50mm lens for this shot, those people in the back row would be getting quite blurred at f/2.8.
Another handy hint is to position the chairs at the front at differing angles and not all pointing straight towards the camera. This helps to avoid those crotch type shots, luckily most of yours are hidden by the kids siting on laps. It also helps to make people more engaged with each other and not look so posed. A second tip is to avoid taking shots like this from your eye level, you need to get up higher so that people have to look slightly upwards at the camera. This is to avoid double chins and everyone who has one will thank you for not showing theirs.
With regards to the overall shot, there are a few darker areas in between people. I tried to increase the light in those areas using the fill light slider in Adobe Camera RAW (ACR) but there is too much digital noise which is a problem with smaller sensor type cameras in low light situations and the fact that the camera has used ISO 800. Fill flash light is the best way to avoid this but if you have used the on camera built in flash, it just isn't powerful enough to work well here. The image is also a bit top heavy where you included the windows and sky at the top at the expense of a couple of people losing their feet. I think pointing the camera down a little more would have helped with this and lit the shadows a little better as well.
I've done a little mod where I adjusted the white balance in ACR, cropped the top to lose the 'Exit' sign and a slight bit off the right side of the image. In Photoshop I then reduced the blues and reds in the background to help make the people more prominent, adjusted the levels to get the whites more white. I then roughly cloned out the chairs from the front and used the dodge tool to lighten the shadows as far as I could. Finally I straightened the crooked background using the warp tool and sharpened the image.
Hope this helps