The tax hike of 3.99% will include a 2% rise to pay for adult social care after falls in government funding.
Unions have said the £18m of cuts being made to the budget was putting staff and services under pressure.
The council says the budget is balanced but critics say the plan lacks specific details of where the cuts will fall.
Council leader John Osman said: "As we move forward the financial picture will get worse before it gets better.
"It's not until round about 2020 that the financial status of the council will improve, according to what the government has told us so far."
He added there would be some "quick wins" of cuts which would not affect frontline services.
The opposing Liberal Democrats are calling for more details about which services will be affected.
Leader of the Liberal Democrat County Group Jane Lock said: "It is a disgrace that the Conservatives will not give councillors and residents any information about the cuts coming on 1st April, yet they are happy to increase council tax yet again."
Independent Mike Rigby said: "I'm very worried because it was very clear last year the budget unravelled, so within three months the council admitted it was £24m overspent.
"I'm worried the same thing will happen again this year, particularly this year because we have no detail about where the cuts will fall."
Taxpayers in Somerset will face at least a £50 rise in council tax after the Conservative-led county council agreed its budget earlier.