All of the financial details that you deal with - you want to know your Town finance director is a straight shooter and knows what (s)he is talking about.
Budgets w/b primarily town spending, but when you get into pensions and shared services 'co-ops' with other towns or state sponsored pooling of purchases, health insurance and what-not, make sure you know that your finance director is experienced. For instance - how much pension cost does town incur? Current vested benefits and/or future obligations. What are assumptions on expected earning on investments - I know up until a few years ago you would say the pension would earn '8%' on their assets - world we are living in today w/b more like 3-4%. That change would double the amount of CASH/TAXES required to fund pensions - current and those of current employees before they retire. You might look at providing benefits like for-profit companies do - defined contribution plans like 401k, vs defined benefits plans which are enormously expensive to taxpayers. If you have a sharp finance director that can outline what the savings w/b to taxpayers - it would be HUGE. Welcome to the real world government employees - just like the 'real world' has been doing for the past 30 years.
Pensions and health insurance items are things that can sneak up and bite you (meaning taxpayers) in the butt in a HUGE way. You want to have a sharp finance director that knows about this stuff and can advise town council in a realistic manner. Our town has outsourced a lot of jobs that used t/b done by town employees - like garbage, parks, street paving - turned out town had high employee turnover, so the generous pay & benefits was going to waste - 'for profit' companies did it much cheaper than town.
You also want to know which other town council members are savvy with finance as well. When you have facts lined up in irrefutable presentation, it is hard to argue 'but being excessively generous to union employees that contribute to my campaigns' are worth sticking it to the hardworking property owner and taxpayer.