You've been threre 2 years that's certainly enough time to give it a chance and time to see what comes up and how it manages/looks etc.
Badly trimmed hedges are better than over grown. And chances are it'll grow back anyhow even if you do make it a bit lopsided. Have you got a hedge trimmer to do these? Invest in some of you don't. So much quicker than chopping with the shears. Just trim over the top and sides, try and keep it level ish but don't worry too much. It will grow back!
Ask the neighbours for the number of the guy you've seen and give him a call. Say you only want an occassionally tidy up of your bigger shrubs and ask what his rate per hour is. The going rate around here varies depending on how professional they are and what the work is. I have a friend who mows my parents grass (2 hours work) for 12 quid cash using their mower. Before that they had a guy come with his own tools, took cuttings away, proper invoice etc (had been to hoticultureal college etc) and he obviously charged a lot more.
If you're not too bothered about the welfare of the plants (and are secretly hoping they'll keel over) go for the cheaper option 
Major pruning should only be a once, possibly twice a year job. And not everything will need doing every year. Do you have somewhere to put/dispose of the trimmings? Some branches may be quite big if it's not been done in a while and he may charge extra to take them away. If you do it yourself you will need loppers, possibly a small garden saw, and time/inclination to do it.
Do you like the roses? Basically prunning helps them to flower more but they will still flower without it. Also allowing air and stopping stems crossing can prevent disease (if you worry about such things - My garden phiolosophy is that it has to look good, look after itself mostly or at least be enjoyable enough to view for the effort required).
Roses are pretty thorny though and nasty to deal with. Apart from climbing roses I'm not that keen on them myself. Yes they look lovely but not worth the effort needed imo.
Would you miss them if they went? Ask around your neighbours to see if any of them are interested in taking any of them off your hands and tell them to come take what they like! That's more of an autumn or very early spring job though so you'll be lumbered with them a bit longer if you go that route unless your neighbours feel brave enough to move them now. Do you know what colour or types they are? If not be honest on that. If they're not in flower say it's pot luck, or if you know one is definitely red or yellow tell them that much.
If you can't rehome them and want them gone ask the gardening guy how much he'd charge to clear the bed for you. Tell him you're not fussy when he does it and he can come and do it any time he has a free day. They will be horrible to take out yourself (you'll have to buy protective gloves and they'll still go through those) so factor that in to the cost of getting someone else to do it. And then grass it over when they're gone.
Same with the trees in the lawn. How big are they? Presumably they wouldn't have a tree preservation order on them....? If fairly small (you can get your hand around the stems) and unlikely to fall on house or fence if they went down, then ask the gardening guy for a quote and bear in mind you will have to dispose of the bits (bonfire?).
If they are much bigger trees then consider the cost and effort of doing it yourself (buying an axe, ropes, or chainsaw hire), possibly it falling on neighbours fence/house... You may need to ask a tree surgeon to do it for you. The plus side is it should be insured so if they break your neighbours fence they have to replace it. And they take the bits away too.
If you do it yourself cut the branches off first and then reduce the trunk from teh top downwards. Wood is much heavier than you realise so take it down in manageable bits.
Get as much to lawn as you can and get those swings up. That's assuming the trees aren't big enough to support a swing from them as is? :)