My dad was a high school French teacher, so I never could get away from learning French. I ended up studying it as a minor through university. I always liked the language. It's also fairly easy, coming from English. Though one is Germanic and the other Romance, both languages follow very similar verb tense structures, and there is of course an enormous amount of vocabulary in English that is drawn from French (thanks to those nasty Normans).
My mother's side is Prussian, so I grew up hearing a lot of German (in a now-extinct Prussian accent!) but I never really learned it until I was able to take it in high school. German vocabulary is a lot to handle.
I never really got a hold on grammar until I studied Latin in my first year of university. Latin was a lot of fun, and it is immeasurably useful. Grammar, vocabulary, sentence structure. You will improve your speaking and writing in your mother tongue.
With my French, I can read a fair bit of Spanish, and because I'd studied German I was able to pick up Yiddish with remarkable ease. (Modern standard German and Yiddish are both descended from Middle High German.) With a background in German, and with a little bit of familiarisation, I can also read basic Dutch, Frisian, Norwegian, Danish, Swedish, Icelandic... As my Yiddish professor said, "All the Germanic languages are basically just dialects."
A year of Latin helped prepare me for Koiné Greek, which I studied for about 4 months. Case systems take some getting used to. I think I was lucky in going from English to German to Latin to Greek. It'd have been a nightmare to attempt Greek having known only English! I figure an early start in basic Greek is helpful if you want to take it up later. Teach your kids to sing the alphabet and have them learn some words. There are a few books out there, but it seems not nearly as much as for Latin.
Hebrew was tricky, since it was my first non-Indo-European language. Took a while to get the hang of it, but it clicked at the end of the term. I can hardly remember a word of it now, but I've already taught my son, now three, to sing the Hebrew alphabet. I intend to review it so I can teach him some basics when he's a bit older.
I studied some Irish Gaelic and Welsh in university as well. Modern Welsh, though not particularly useful (since pretty much all the native speakers today also speak English) is very easy to learn. Just remember to keep hydrated and carry a handkerchief.
Korean is hard. If there is a language opposite to English, Korean is it. Granted, I have not really applied myself to it, and I'm a little older than I used to be.
If I had the time, I'd study Old English and Middle English.