If you want to say you've hiked the highest mountain in a European country, there may be no easier peak to top than Kékes (also known as Kékestető due to the large television tower at the top). At 1,014 m, the highest spot in Hungary may not be the most dizzying summit you'll tackle in your lifetime, but it certainly makes for a surprisingly enjoyable day of hiking. Plus, if you don't mind paying the roughly 500 HUF ($2), you can take the elevator to the lookout walkway on the TV tower for some fantastic views over the rest of Mátra mountains and the north-central Hungarian countryside. This is actually highly recommended, since the views from the summit itself are actually fairly lackluster due to the parking lot and trees.
To get to Kékes, you'll either need to rent a car from Budapest or hop a bus headed from Puskas Ferenc Stadion (on line 2 of the metro) headed towards Mátrafüred. There's a stop right at the start of the Mátrafüred trail head, which is a 5.5 km 1-way trek, although if you see the included link you can tackle the hike from several other trails as well. From here, just follow the well-marked trail (either the blue cross or the blue triangle) and work your way to the top of the mountain. The path varies between fairly easy and somewhat strenuous inclines, however with a little water and a few breathers the route can be tackled by any capable hiker. We had the trail entirely to ourselves when we hiked it, which made for a fantastic experience.
Note:There's a small ski hill and lift at the top of the summit, so it's safe to assume that you can ski/snowboard here in the winter as well. Also, while you can conceivably do the round trip travel and hike in one day from the capital, I'd highly recommend pairing it with a visit to more less-frequented destinations like the beautiful town of Eger.