By ‘government relations’, do you mean communications?
I have worked in the civil service, not any more thank God, but the policy people I met had usually come in as graduate entrants and chosen to specialise in that area. Not many people joined that discipline externally from non-public sector policy roles. Policy can be quite cerebral compared to other roles and the people working in that area tended to fit a certain mould: serious, and took themselves very seriously.
Comms was/is easier to break into, because people from non-public sector backgrounds were considered. If that is what you are interested in, look for entry level corporate or (similar but less useful) internal comms roles.
I'd be interested in policy jobs both in and outside the Civil Service. E.g. policy jobs in charities.
For government relations, I mean roles with private sector/public sector organisations and charities that influence Parliament and improve government policy.
For instance, I saw a Policy Officer job with Teach First advertised earlier this month, but didn't go for it as I don't have relevant experience. Does anyone have any tips on what would be the best way of getting into this sort of job?
Bombardier I did apply for the Fast Stream earlier this year, but didn't get past the Etray exercise unfortunately.
CharityJobs is quite good, weekly job listing. I started my grad job in consulting but working on government contracts so got policy experience drafting papers that way (I failed FastStream). If you are only starting out look at every grad scheme going. I had a v good local government job and lots of our junior policy staff were brought in on temp contracts - have to chat to temp agencies and see which one in your area has a contract with your local government body? Once you get a foot in the door you can apply for internal role - I know policy staff in DEFRA/ Treasury and Decc that all started on temp contracts through agency. Think about other agencies too - Often has some great roles, and policy skills are V transferable.
Citizen Advice experience is great on CV - maybe look into advocacy roles with charities? It's a great way in.
And in terms of skill set look at making sure you have sharp numerical and analytical skills, project management and excellent written, notetaking skills - lots of which you can learn from online courses.