Total health spending is cut by 7.6% in real terms, with the total spending on education and local government falling by a similar amount.
Hardest-hit in the draft budget are economy and transport and environment, down by more than 21% over three years.
Budget Minister Jane Hutt said it was in the toughest era since devolution.
Ministers said they have been able to give small cash increases in the day-to-day funding for the delivery of health services and for schools.
Spending on big infrastructure projects like building roads, hospitals and schools are likely to be cut dramatically over the coming years, with many planned schemes delayed or cancelled altogether.
The capital budget for the NHS has taken a severe hit, falling from £283.3m this year to £205.2m in 2013-14.
The dramatic fall will mean that it is unlikely that any major new hospitals can be built for many years, since maintenance of existing NHS buildings and equipment will also have to come out of this budget.
One of the main pledges from the assembly government was that the draft budget would deliver cash increases for schools and social services over the next three years.
They say revenue funding for social services delivered by local government will rise from £1.056bn to £1.08bn by 2013-14, and funding for schools will increase from £1.83bn to £1.9bn over the same period.
However, both of these funding streams are currently paid direct to councils via the Revenue Support Grant (RSG) and, at present, local authorities do not have to spend all of this funding directly on schools or social services.
An assembly government source said that ministers were examining ways of "passporting" this money so that it had to be spent as intended in its entirety.
Unveiling the draft budget Ms Hutt said the assembly government was on record as stating its opposition to the speed and depth of cuts imposed by the UK government at a crucial phase of recovery from recession.
"However having been dealt this hand, we are determined to do what it takes to stand up for the people of Wales and continue to provide the right support for those who need it most," she said.
"Our draft budget is about building resilience - resilience in our economy, and in the provision of vital services on which people depend."
The minister said that with the budget next year due to fall by £860m - it will be £1.8bn lower by 2014-15 - difficult choices had had to be made.
"Clearly there have had to be reductions in some activities," she added.
"However, we have approached these choices on the basis of the best evidence available and have worked hard to reduce the duration and impact of any reductions."
The Labour-Plaid Cymru coalition government's draft budget met heavy criticism from its political opponents.
The Conservatives' shadow finance minister Nick Ramsay described it as "completely unacceptable", saying it put frontline services at risk.
Kirsty Williams, leader of the Welsh Liberal Democrats, described the draft budget as a missed opportunity to build Wales' economic future, root out waste in the NHS and target funding at the poorest school children.
However Plaid Cymru's finance spokesperson Chris Franks praised the assembly government for its refusal to follow Tory plans to cut budgets by up to a quarter.