Day Hikes near Berlin

Amy, I'm responding because I've been to the Spreewald, and probably not too many others here have--not because I can be very helpful (sorry).

The Spreewald is a marshy area that I remember as being very flat. I primarily took one of the boat trips, just walking around for 15 minutes or so at one of the stops. Plus the walk from the train station to the historic area is of some consequence (I'd guess about a mile; zoom in on Google map of Lubbenau). I enjoyed that day, but I don't think it's what you're looking for--at least not the part of the Spreewald I visited.

I just looked at the area around Berlin on Google Maps, and I see "Naturepark Westhavelland" and "Naturpark Maerkische Schweiz", plus quite a lot of places with "Wald" in their names--which I think means "forest". Surely someone on the forum can point you in a good direction. If not, I'd ask the Berlin tourist office folks for a recommendation. The Germans are big-time hikers.

Edited to add: I Googled "hiking Spreewald" and got a bunch of hits. You can find information online about the walking trails. I'm just doubtful about elevation gain. Oh--I don't remember mosquitoes, but I'd think they might be an issue if they find you attractive.

Berlin lies in the Northern German lowlands, which are flat as a pancake. To find a hike with a elevation change of 1000ft you first have to travel for about 3 hours either to the west or south.
Hiking in the Spreewald is beautiful, but it's something completely different. Forrests, meadows and waterways, cute little villages, if you are lucky kingfishers or dippers, but no grand vistas.

Is there a map anywhere of the hiking trails? Are the hiking trails
the hilly-dirt-trail-through-the-woods kind or are they the
paved-level-walking-trail-along-a-river kind?

Typical hiking trail
Hiking maps exist of course, but they usually also show cycle paths and waterways. And for just one day I surely wouldn't buy one. The area is very touristy and well-marked, and printing out a online map, or a offline map on a smartphone, should be enough. You would probably start in Lübbenau, and the most popular hikes from there are the Sorbian villages of Lehde and Leipe, and a popular restaurant called Wotschofska, which are almost impossible to miss.

But I would recommend something different: how about Potsdam? Potsdam is surrounded by beautiful parks and lakes. Most foreign tourists usually just visit Sanssouci palace, but the park is HUGE. If you want to see everything... New Palace, Chinese Tea House, Roman Baths, Ruin mountain, Belvedere and much more.... you need a full day, and you would walk ~10km or so.
Here is a list of all structures in the park:
Buildings in Sanssouci Park

Or see the other parks of the city. Last year I did this walking tour:
From Babelsberg to Belvedere on Pfingsberg and back

Take the bus from Berlin to Glienicke Palace and cross the famous Spy Bridge. Next is the northern end of the Berliner Vorstadt, the poshest part of the city where Germanys rich and famous live, with a reconstructed royal pier called "Kongnaes". Then you cross the New Garden with Cicilienhof Palace and hike up to Belvedere on the Pfingstberg. From there back to the New Garden, via the Humboldt bridge to Babelsberg Park with Babelsberg Palace . Before you return to Glienicke Palace you cross a cute little village with Swiss cottages, a hunting castle and a very good ice cream parlor.

The state around Berlin is named Brandenburg. The tourism website lists some Brandenburg hiking routes.

Be aware that neither Berlin nor Brandenburg are flat like a pancake. This area you will find around Bremen and along North Sea coast closed to Wadden Sea. All other northern German areas have low and long hills (may feel flat) shaped by glacial period. Hoher Fläming Nature Park and Maerkische Schweiz Nature Park are nice examples for that around Berlin.

VisitBerlin has a good overview of daytrips around Berlin from which some include hiking.

Hope that helps a little bit.

Should you want to stay in Berlin to do this hiking, how about Grunewald? As suggested, I would also recommend the Potsdam and Babelsberg area. A good number of the famous Potsdamer Schlösser can be seen.