Mincha before candle-lighting, and candles that need to burn far longer than usual.
Q. What’s done differently when Chanuka falls on Shabbat?
Firstly, it’s preferable on Fri. afternoon, to daven mincha before lighting Chanuka candles, because mincha “belongs” to Fri. whereas the Chanuka candles already “belong” to Shabbat. To this end, most communities have early minyanim for mincha, so that people don’t need to hurry back home afterwards to light. If it’s complicated, then you can light candles before going to shul. In addition, one must use longer/thicker candles or more oil than usual (the regular ones last about 40 minutes, and even less on the 8th night when the many candles, add abundant heat!). The candles or oil must be long enough to burn a half hour “into” your proper time. Accordingly, in Israel, for those who’s “time” is usually to light at sunset (mostly Ashkenazim), theirs must burn almost an hour (22-24 minutes before sunset+ 30 minutes of burning time), and for Jerusalemites, at least an hour+ 10 minutes, (for they light Shabbat candles 40 minutes before sunset). For sfaradim and those who light from night time (15-20 minutes later than sunset), their menorah’s must last that much longer!
R. Moshe Feinstein adds, that although candles should usually be basically the same height (so viewers can tell which day of Chanuka it is), since viewers know that Shabbat is special and different, if you don’t have enough long/fat candles, you can light some longer and some shorter ones, together, and if there’s no choice, some which will last the full time and others that won’t.