Deffo buy imo. We bought our first home about 4 years ago and never regretted it! Best thing is we have filled it with pets!
We practically lived together for a while but still rented full time together for about six months first, worthwhile move I think to get used to sharing all the thing that you might not have done before e.g. finances/shopping/housework etc.
I'd personally steer clear of those shared ownership things - they are often flats (I'd never buy a flat, being at the mercy of some property management company to look after the building/grounds etc? No thanks! Had enough bad experiences of that while renting!) and I looked into them a while back and by the time you'd paid the mortgage, the rent on the part you don't own and the ground rent, it worked out more than just buying a house with garden and drive somewhere a bit further out. If you do consider going down that route just make sure you work out the sums carefully!