Mei Tei slings are suitable for newborns, but you need to be aware that cheap ones can be made from unsuitable material that isn't supportive enough, especially as they get bigger and heavier. I never got on with Mei Teis but that was just me - I prefered wraps as they are nice and snuggly!
Have you had a look at slingguide.co.uk/ - there is loads of advice and information on there.
Personally I prefer wraps, I used a stretchy (Moby) for DS2 and will be getting it out again in 4 weeks for newborn. DH doesn't like wraps so I also got a second hand Patapum for him to use. You can get second hand stretchies for about £20 and I paid £15 for the patapum, also second hand.
Once DS2 got a bit bigger I started on woven wraps which I will also use for this new one as well.
Have you looked for second hand wraps / slings? Not sure if I am allowed to post link to the forum which I found loads on, so I will PM it to you once I have figured out how to PM!