I had a similar experience to you, I went to an electrologist and she said that since I have fair skin and dark hair that I am a good candidate for laser first.
The key with laser is that you have to make sure that the laser is set aggressively! I did laser a few years ago and the settings just weren't high enough to change much. The hair slowly fell out, but it grew back.
Recently I've gone to a new laser place, and they are using much more aggressive settings. I take ice packs along with me so that I can immediately ice my face down (especially my upper lip and chin areas). The hair has responded MUCH better this time, and I feel like I'm actually making progress. The hairs are starting to come back as white now so probably after a few more treatments I'll just move to electrolysis.
If you do laser first, just make sure you have them turn up the machine as high as possible without doing skin damage. Laser places tend to keep the settings low because it means they make more money when you come in for more sessions.
I'm using an Alexandrite laser with 18mm spot size and 16J energy setting, but I'd like to move to 18J.