There are no major differences between the Sephardic and the Ashkenazi traditions in the order of Ḥanuka candle-lighting, just a few minor variations.
Some of them are:
1. The Ashkenazi tradition is to say the Berakha: "lehadliq ner shel Ḥanuka", while Sepharadim say: "lehadliq ner Ḥanuka", without the word "shel." It is interesting to know that, although there is not semantic difference between the two versions, the original version of the berakha (as per Maimonides; see MT, Ḥanuka 3:4 ) was "lehadliq ner shel Ḥanuka".
2. In the Ashkenazi Minhag, the auxiliary candle (shamash) is lit first 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 shamashis seen as auxiliary because it avoids benefiting from the light of the candles, not necessarily for lighting with it the other candles.
3. In most Sephardic communities, it is customary to light only one Ḥanukia for all members of the family. In many Ashkenazi communities the custom is to light one Ḥanukia for each member of the family. Following the Ashkenazi tradition, for example, a student who lives in his own apartment would light his or her own Ḥanukia with Berakha (even if he is still depending on his parents). Incidentally, this is also the case regarding Shabbat candles: while according to the Sephardic Minhag only the mother would light the Shabbat candles, in the Ashkenazi Minhag the daughters also light their own candles, saying Berakha for it.
4. Playing with the Dreidel, spinner or sebibon is originally an Ashkenazi custom, that Sepharadim did not use to practice in the past. Same as Ḥanuka Gelt (money or gifts to the children).
Obviously, in these matters there is no right or wrong. Each person should follow his community's and family's traditions.