I'm a vegetarian of many years. It's kind of what you make of it. You can be a healthy or an unhealthy vegetarian, just like you can be a healthy or unhealthy meat-eater. I am overweight, but otherwise quite healthy with regards to cholesterol, sugars, etc. Whether that is due to the veg diet, I don't know.

When I have been at my best weight, it was because I was careful not to let too many carbs into my diet and focused on including a variety of proteins in my diet. There are some reports that there can be drawbacks to having too much soy in the diet (and just as many reports that a lot of soy is okay) so seeking a balance of different proteins is always a good idea.

If you aren't doing it for moral reasons, then your idea of first cutting out red meat makes good sense. If you like having fewer animal proteins in your diet, you could cut out chicken after that. I think switching over from meat can be a bit of an effort if one is not used to thinking meatless!

Good luck!