I probably saw the first two seasons of Falling Skies. It's okay I guess but it's not something that I would recommend. There are just so many better shows out there. Last ship was pretty solid. I'd go with that because they are only one season in and it starts back up soon.
For a light summer show I liked both. Falling Skies was good for the first season then, imo went down hill quickly in the second and I dropped it. Last ship will probably go that way too, but I liked the first season quite a bit for what it is.
I was frustrated with the end of S1 of the Last Ship. I don't like where they are going with it. Needs more Russians and ship stuff.
TBH, I forget what happens in these shows almost as soon as the season is over. I couldn't tell you how they left things off. Really depend on the "last season on..." Tyrant was probably my fave of the "summer fluff' shows for lack of a better term