I am traveling to Finland this fall. The people I'm visiting say there's no mountains, but a lot of good hiking and some rock climbing. Can anyone suggest an interesting area where I can get a mixture of hiking, backpacking, and rock climbing?

I was there about 15 yrs. ago and saw little rock. Some really random erratics but not much else. I know there is some rock climbing there but there's a lot more the further west you head. Sweden has a decent amount and Norway an ungodly amount. Nice hiking but it's mostly going to be birch forests unless you head to the far north. That looked really interesting but I didn't have time to venture up there on my trip. Also, as you get further north above the Arctic Circle you get closer to and more similar geography to those northern, more mountainous and rockier portions of Sweden and Norway. Definitely check it out but also be open to a nice canoe trip (there are LOTS of those) and maybe even a ferry ride to Tallin (in Estonia) or a quick train ride into St. Petersburg. That last thing may have been the highlight of my trip.

Two Fins go to a cabin deep in the forest in the middle of winter.

The cabin is empty except for a table, two chairs, a bottle of ice-cold vodka, and two shot glasses. There is a toasty fire in the fireplace.

The two Fins sit down at the table across from one another. One pours each of them a glass of vodka. They raise their glasses, one of them toasts "Kippis!" (Cheers!), and they gulp the vodka.

One of the Fins pours each of them another shot. They raise their glasses, one of them toasts "Kippis!" and they gulp the vodka.

Again, one of the Fins pours each of them another shot of vodka. They raise their glasses and one of them toasts "Kippis!"

The other Fin gets angry and shouts, "HEY! ARE YOU GONNA TALK -- OR ARE YOU GONNA DRINK?"

Fall should be a nice time of year. I was there in the middle of summer and the flies were unbelievable. I would have traded even a heavy cloud of Sierra mosquitos for those voracious beasts.

When exactly are you going? Autumn is long period and depending in which end makes a difference. After the autumnal equinox the 22 September, the days becomes short pretty fast. The further north, the faster it goes. When the snow comes it doesn't matter all that much, but the period of short days and no snow is really dark.
Otherwise, the arctic north is way coolest. South of Finland is pretty much as... maybe Maine? A beautiful place is Punkaharju. It's located in the southeast.