Finance Minister Jane Hutt told AMs the 2016/17 draft budget backed Labour's "priorities for Wales and its future".
Total health spending rises by 4.1% while education and skills goes up by 1.1%, but local government faces a cut of 2%.
The Conservatives said the spending plans offered "too little, far too late for our hardworking NHS staff".
Ms Hutt said the Welsh government had faced a 3.6% cut in UK government funding over five years, once the effects of inflation were taken into account.
About £260m will go on day-to-day, or revenue, funding for the NHS and another £33m will be spent on infrastructure, maintaining the NHS estate and equipment.
There will, however, be a reduction of £15m from other parts of the health budget, giving a net figure for the NHS of £278m.
Although the education and skills funding is getting an increase, the body which funds Welsh universities said its budget was being cut by £41m, nearly a third of the total.
The Higher Education Funding Council for Wales had already warned not enough money was available to Welsh universities, because of the way tuition fees are funded.
Boosts to the Communities, Economy and Natural Resources budgets are largely made up of capital spending increases, for example housing and flood defence projects.
Their day-to-day, or revenue, budgets, are being cut.
Ms Hutt said her aim was to protect the services that mattered most to people, within the financial constraints.
"It has been another challenging settlement which has been set against the backdrop of successive real terms cuts to our Budget over the last five years," she said.
"We have continued our record investment in health with more than a quarter of a billion pounds going to the Welsh NHS in 2016-17 - demonstrating our wider approach to the health and social care and the value of preventative spend."
The lion's share of the Welsh government's total expenditure of £15.9bn comes from the UK Treasury, apart from about £1bn raised in business rates.
Almost half of the budget is spent on health, which has seen above-inflation increases since October 2013, when the Welsh government reversed its controversial decision not to protect the NHS budget between 2011 and 2013.
Conservative Shadow Finance Minister Nick Ramsay said Labour's latest budget boost for health could not make up for previous cuts to NHS funding.
"Hospital downgrading, huge delays in waiting times, a failure to recruit staff; this chaos is a direct result of Labour's record-breaking NHS budget cuts," he said.
"By failing to protect the budget, our health service has been starved of £1bn since 2010/11."
However, Plaid Cymru AM Alun Ffred Jones "warmly welcomed" the extra NHS cash.
"I do very much hope that this will enable waiting lists to be substantially reduced, in light of the harsh criticism of the Royal College of Surgeons in their statement yesterday," he said.
Liberal Democrat Peter Black highlighted an extra £1,150 each school would receive for every pupil eligible for free school meals, as part of a two year deal his party had struck with Labour ministers in 2014.
But he sought assurances that extra NHS money would not be used simply to pay off health board deficits, rather than to invest in the service.

Airbus UK Broughton's Ryan Wade, 28, was banned for three "approved competition" matches from 21 April.
A hearing on 5 April found he "brought the game of association football or the association into disrepute".
Johnson, 28, was jailed for six years for grooming and sexual activity with a girl aged 15.
The former Sunderland player has lodged an appeal against his sentence.
BBC Wales understands the nature of the tweets was deemed offensive and were sympathetic towards Johnson.
The Welsh Premier League said the three-man panel sat at the headquarters of the Football Association of Wales.
Wade has been suspended from all football-related activity and fined Â£150, as well as a Â£50 contribution towards costs.
The club, based in Broughton, Flintshire, said in a statement: "The club carried out our own investigation and disciplinary process at the time and that has now concluded."
Wade signed for Airbus almost four years ago and has made 28 appearances this season, scoring four times.
The NHS is to receive a net boost of £278m under the Welsh government's spending plans.
[NEXT_CONCEPT]
A Welsh Premier League footballer has been banned for tweets he posted after the trial of ex-England international Adam Johnson.