How to move all clips to the left after I cut section out?

Hi all,

Know really simple question, but struggling to find a simple answer. 

I've got lots of video clips that I'm putting in the Premiere timeline and, as I'm now editing them, I'm frequently having to cut out sections and then shift select every clip to the right of it and drag it to the left to fill the 'gap'.

Pretty sure there'll be a much smarter way to do this from a pro....!

Whilst I'm asking this, I quite often have the reverse - ie I realise I need to insert a video somewhere and need to move all the clips to the right. It's annoyingly been overwriting existing video clips if I drag them into the timeline without any warning message.

Any tips very welcome!