In the more recent games, there are NPCs that you can talk to and they will tell you what type of Hidden Power your Pokémon has. According to Bulbapedia, the locations of the NPCs are:
- Platinum: Veilstone Game Corner
- Heart Gold/Soul Silver: Celadon Game Corner
- Black/White: Mistralton City
- XY: Anistar City
- South-Eastern-most house
- Also the location of the Hidden Power TM.
- Omega Ruby/Alpha Sapphire: Fortree City
- Top row of houses, second from the left.
- Also the location of the Hidden Power TM
If you want to find out through combat, here's something I threw together (Disclaimer: there's likely a much more efficient way to figure it out):
1. Battle a Patrat / Lillipup.
Super effective: Fighting. Didn't affect: Ghost. Normal effectiveness: goto 2
2. Battle a Cottonee / Petilil.
Super effective: goto 3. Not very effective: goto 4. Normal effectiveness: goto 5.
3. Battle a Klink.
Super effective: Fire. Not very effective: goto 6. Didn't affect: Poison.
4. Battle a Roggenrola.
Super effective: goto 7. Normal effectiveness: Electric.
5. Battle a Sewaddle.
Super effective: Rock. Normal effectiveness: goto 8.
6. Battle a Woobat.
Super effective: goto 9. Normal effectiveness: goto 10.
7. Battle a Ducklett.
Normal effectiveness: Grass. Not very effective: Water. Didn't affect: Ground.
8. Battle a Sandile.
Normal effectiveness: goto 11. Not very effective: Dark. Didn't affect: Psychic.
9. Battle a Ferroseed.
Normal effectiveness: Ice. Not very effective: goto 12.
10. Battle a Timburr.
Super effective: Flying. Normal effectiveness: goto 13. Not very effective: Bug.
11. Battle a Darumaka.
Normal effectiveness: goto 14. Not very effective: Steel.
12. Battle a Drilbur.
Normal effectiveness: Dark. Not very effective: Rock.
13. Battle a Trubbish.
Normal effectiveness: goto 15. Not very effective: Poison.
14. Battle a Deino.
Super effective: Dragon. Normal effectiveness: Normal. Not very effective: Dark. Didn't affect: Psychic.
15. Battle a Blitzle.
Normal effectiveness: Dragon. Not very effective: Steel.
My thought process (possible types at each stage) is in the revision history of this answer.
In Gen4 and Gen5, there's no distinction between an unnamed Pokémon and one nicknamed its species name. This differs from Gen3, which had a "nicknamed" flag. A Pokémon loses its nickname when it evolves, if its nickname is the same as its species name in any language.
In one case, I had a Japanese Vibrava (named ビブラーバ, its species name). I had to evolve it to Flygon to fill a gap in my Pokédex, but I didn't want to because I knew it'd lose the nickname. In the end, though, I decided to let Flygons be Flygons.
Best Answer
Generation V removed contests, and therefore provides no methods to raise or check any of Contest stats of a Pokémon, excluding the Feebas evolution for checking Beauty.
There are a few approaches involving third-party tools to viewing the stats of your Pokémon in detail (such as dumping the save, or tricking the game to use an incorrect GTS server), and since the old contest stats are still present on a Pokémon (which is why the classic Feebas evolution still works), you can use one of those to view these stats. However, they have no effect at all in Generation V (again, excluding Feebas evolution), so there's not much point in doing so.