You've already gotten quite a few good answers, but most of them are focusing on the mountains to the north-east of the city center. I'll try to give a more thorough answer, including all of the 7 mountains, and how to reach them.
First, let's look at a list of the 7 mountains:
The 7 Mountains:
- Lyderhorn (396 MAMSL*)
- Damsgårdsfjellet (317 MAMSL)
- Løvstakken (477 MAMSL)
- Ulriken (643 MAMSL)
- Fløyen (400 MAMSL)
- Rundemannen (568 MAMSL)
- Sandviksfjellet (417 MAMSL)
*MAMSL: Meters Above (Mean) Sea Level
The 7 Mountains Hike
The most obvious way to reach all 7 mountains, is doing "The 7-Mountains Hike" (7-fjellsturen). This is a 34 km hike, with a combined accent of 2400 meters. This is a challenging trip that will take approximately 8 hours for an adult man in good physical condition and not including any breaks. This guy did it in less than 4 hours, but note that the headline in the article is: "Hardly particularly healthy". As you probably realize, this is a trip you should be well prepared for, but it's not unfeasible. Each May there's an organized hike where more than 8000 participate, and most finishes.
You won't see many people carrying maps, and on a sunny day it shouldn't be necessary either. However, most people you'll see up there have been hiking on these mountains many times before and knows the trails well. You will also likely be among very few going the entire 7 Mountains Hike if you choose a random day.
Gravdal, the starting point is a 15 minutes bus ride from the city center, and the bus goes approximately every 30 minutes.
Suppose you don't want to do the entire trip, there's the possibility of only doing the "4 Mountain Hike", which is only the 4 last mountains. If you do so, you may take the bus up to where you start your hike to Ulriken (the bus stop is called Montana). (To nitpickers: Yes, the official 4 Mountain Hike starts by Årstad VGS, but it's easier to find, and easier to hike from Montana.) More info below.
There are signs telling you which way to go, but sometimes these may be hard to spot. If you're by the road, green signs like the one below show you which way to go to find the starting point of the hike. There are such signs for all of the mountains, but you need to be in the right spot to begin with. The tourist information can tell you where to go.
Foto: John Hassal (http://www.hassall.no/index.php?pageId=galleri&evtId=369)
Once you've started your hike, you'll see different signs, such as the ones below. As you can see, they don't show you how long the hike is, only the direction. You also have signs on the way telling you if you're on the right track.
Unless you're determined to do the 7 Mountain Hike, your time is probably best spent walking over Vidden. That's a hike going from Ulriken to Fløyen and offers great views over the city and its surroundings. You can shorten the hike by taking Fløibanen and/or Ulriksbanen. If you walk up from Montana to Ulriken over the plateau to Fløyen, it will probably take you about 5 hours. Most of the path is over rocky terrain, but there are a couple of sections where it can be quite wet and muddy, so waterproof shoes are sensible.
2: Pick and choose
If yoy're in Bergen, visiting Fløyen is an absolute must! If you choose to walk or take the funicular is up to you. You can walk from Fløyen to Rundemannen quite easily, but in my opinion that's not worth the trip. (You'll pass Rundemannen if you walk Vidden).
The hike to Sandviksfjellet is also simple, and might be a nice trip if you want to walk around a bit. You won't risk getting any dirt on your shoes if you go to Sandviksfjellet, as the paths around Fløyen all look like the ones below (that's actually the path from Fløyen to Sandviksfjellet and not just a random one).
Going up and down Ulriken is also a nice trip, and you can take the cable car one of the ways. The view is great, and there's a restaurant etc. on the top. It's hard to go wrong once you get to Montana, and that's a 20 minutes bus ride from the city center.
My personal favorite is Løvstakken, which has a great 360 degrees panoramic view. From the city center it's easiest to take the bus to Solheimslien and follow the signs and path. Also this path is hard to miss. If you're only going up and down Løvstakken you don't need maps/compass. As with Vidden, it might be a bit muddy if it's been raining.
Shoe wear: I always use running shoes, regardless of which of the mountains / hikes I'm doing. There's a big chance it'll be wet, but I find hiking boots to be a bit overkill.