I've currently two rescue cats + one that adopted us (malnourished, flea-ridden and aged about 6 weeks) plus a re-homed dog (advertised on an expat site over here) who arrived with mental baggage such as separation anxiety. With time and patience all his problems were resolved in a couple of months. Of course, not all animals come with baggage but he'd probably have been put down if we hadn't taken him

and he's turned out to be the best trained dog we've ever had. No thanks to us!
Assess the rescue and make sure
they're clean and efficient, know what they're doing and the animals are in a routine and as stress-free as possible. We adopted two kittens from a very poor rescue who arrived with ringworm which proved expensive and time-consuming to eradicate.
You should expect - and welcome - being assessed by the rescue themlselves. I would always want to take an animal that had been neutered or spayed and for me, I'd always want one that had been assessed to be pretty bomb-proof around people - you can never be 100% sure of any dog, even one you've raised.
This isn't being picky - there are thousands of dogs out there who have had an unlucky life but not been mistreated and they will make excellent companions.
I'd say don't expect to bond instantaneously. You might, but it may take a week or two or a month and initially at least, I think establishing a routine is extremely important to how well the dog settles with you.
And finally, it's not wrong for a rescue to ask for a decent donation. Some people - and I'm
not saying you are like this - tend to think that an animal from a rescue should be free. Well, I think taking on an animal is a responsibility, there may be vet fees in the future so it's not wrong for a rescue to look for an acknowledgement of that possibility. And any donation isn't a payment for your animal but a contribution towards future treatment, sterilising and bed and board for others.
I've made a conscious decision to stick with rescues from now on. I love cats but hate their killing ways so I've decided that when the current batch go the way of all pets (hopefully not for years yet) I'll take on elderly moggies (10 years +) who, I hope, will be more interested in sleeping in the sun on a comfy lap than being killing machines. People often don't want the oldies... but they are often the more rewarding as they're more interested in cuddles and comfort.