A two-hour walkout on Friday by the examiners will be followed by a week of short strikes in the Maritime and Coastguard Agency (MCA).
The Public and Commercial Services union said more will be held throughout June at the Department for Transport (DfT) head office and several agencies.
The DfT said contingency plans were in place to keep vital services running.
The PCS union said the planned closure of Driver and Vehicle Licensing Agency (DVLA) offices would "wipe out the high quality face-to-face services these offices provide to the public and motor traders".
The closure of eight of the UK's 18 coastguard stations would mean "the loss of life-saving local knowledge", it added.
But the government said the changes to its coastguard provision would modernise the service.
PCS general secretary Mark Serwotka said: "With unemployment high and our communities suffering as a result of the government's cuts, ministers should be providing all the help they can to local economies that are crying out for support and investment.
"Instead they're ploughing on with entirely unnecessary and unpopular cuts to vital transport services.
"These strikes across the whole range of transport services form a major part of our ongoing fight against cuts to pensions, jobs and pay."
The PCS said more than half of its members had backed walkouts with four out of five supporting industrial action short of a strike.
Mark Peacock, head of BSM driving school, said: "Taking your driving test is a big day and can be nerve-wracking enough without the thought of strike disruption.
"However, candidates who have chosen to keep their test booking for the afternoon of 8 June should still attend their test so it can go ahead if possible."
The DfT said it was "disappointed" at the decision to strike while it was still in discussions with the union.
The MCA said those striking would be staff responsible for co-ordinating rescues and "the rescuers themselves will continue to operate as normal".
Helicopters would continue to fly and the RNLI would still send out its lifeboats, a spokeswoman added.
Last week, HM Revenue and Customs (HMRC) staff voted to take industrial action over jobs and privatisation after a third of workers took part in the ballot.
The PCS represents 55,000 members in HMRC, including tax advisers and call centre staff.
The union says thousands of jobs face the axe over the next few years on top of 30,000 posts already lost since 2005.
PCS members were involved in a 24-hour UK-wide strike by public sector workers last month over government changes to pensions.
The union is one of the largest in the UK with around 250,000 public sector members.
Driving examiners and coastguards are to hold several strikes against plans they say will hit jobs and pensions.