There are no major differences between the Sephardic and the Ashkenazi traditions in the order of candle lighting, just a few minor variations.
Some of them are:
1. The Ashkenazi tradition is to say the Berakha: Lehadlik ner shel Chanukka, while Sepharadim say: Lehadlik ner Chanukka, without the word "shel." (In our community, however, some families still add the word 'shel' following the Bene Tzion Siddur).
2. In the Ashkenazi Minhag, one first lights the auxiliary candle (Shamash) and with it one lights the rest of the candles. The Sephardic Minhag is to light all the candles first, with a regular match or candle, and the Shamash is lit at the end. In this case, the Shamash is seen as auxiliary, in that it avoids benefiting from the light of the candles, not necessarily for lighting with it the other candles.
3. For most Sepharadim, it is customary to light only one Chanukkia for all members of the family. In many Ashkenazi communities they light one Chanukkia for each member of the family. Following the Ashkenazi tradition, for example, a student who lives in his own apartment, lights his or her own Chanukkia with Berakha, even if he is still dependent on her parents (see here). Incidentally, this is also the case regarding Shabbat candles: while according to the Sephardic Minhag only the mother lights the candles, in the Ashkenazi Minhag the daughters also light their own candle, saying Berakha for it.
4. Playing with the Dreidel, spinner or sebibon is originally an Ashkenazi custom, which Sepharadim did not use to practice in the past. Same as Chanukka Gelt (money or gifts to the children).
Obviously, in these matters there is no right or wrong. Each one should follow his community and family's traditions.
