Advice on trees on our front yard

We moved into this house over a year ago and inherited both these trees. I’m unsure what species they are (can anyone identify?). This is my first home so I’ve never dealt with trees before.

The one on the left have shoots/branches that grow upright really high (it’s almost as tall as our 2-storey house now). Would it be causing any issue or should it be pruned regularly?

The one on the right have new shoots growing up from near the base of the trunk. I’m planning to take them all off and have it grow upwards instead. That’s not a bad thing to do, is it?

I’ve got a local tree surgeon coming over tomorrow for other smaller jobs, just wondering if I should get him to quote for other stuff while he’s here.

Thanks in advance!