Most people dislike the strays that turn up but I want to try and add them to my families. Unfortunately they dont seem to want to visit or I only get the same one. Is there any way to encourage them to visit more often.

I like strays, most of my household pets were once strays. The best way is to adopt them in the first time they visit. The way to make that happen before they leave is alternate what you do, so play, love and also after a couple of seconds of starting the action cancel it. This will make the action very short but still give you the relationship points. To encourage pets leave out food for both and a dog chew toy for dogs.

You can also invite them to parties and, in my experience, they always come when invited.. It's ridiculous but...Hey, it works. They won't wander off the lot until the party's over.

So far I've only had one stray, a dog named Buffy, who strolled onto the Specter estate right after Olive died. I'd planned on getting Nervous a dog anyway, so this worked to my advantage. Nerv befriended and adopted her immediately

You can get the mod Allow Strays on Community Lots.

Strays are fun, until they start digging up your flower beds. I like the concept of strays, I just don't like how wild they are.

Adopting them is hard though, since by the time you're starting to approach friends they get bored/tired and leave and don't care to come back to that lot again :P

This is exactly what I have been experiencing..so i wonder how Jo does it!

Usually the first thing I do after greeting them is give them a treat. Then stroke them, play with them, give them another treat, and repeat, performing different actions when they become available. This way works about 7/10 times for me...

Fast. You alternate between play and give love and you cancel it shorty after they start the action. The reason they leave is you have taken too long to get to the point of being friends. Some actions like give treat don't get canceled, but toss in the air, cat teaser, cancel it a few seconds after they start doing it. Not to say I am always successful, but that's how I get the majority of my pets and it works 90% of the time.

Strays aren't always annoying. One time, one of my Sim's enemies showed up at his apartment lot, not sure for what, but definitely for no good. I was about to have my Sim shoo him off when a stray dog ran up to him and growled. It was hilarious just watching this stray dog giving your worst enemy a good scare!

the thing with visiting strays is to alternate between play, love and treats. I usually give in that order, using higher valued actions as they they become available and constantly checking for the adoption option. In my experience you can build up the friendship fast enough to adopt on the first visit in many if not most cases. I've never got a visiting stray to return for a second attempt.

Create a 'Pet Park' community lot. On the lot have 3 pet bowls, beds, bones, and cat mice. Etc.-everything pertaining to pets. Have 3 or 4 benches behind the phone booth. An angry pet, not cared for may tear up the bench, then you can work with said pet.
Place the pet bowls near - about 6-8 tiles near the front of the lot in order that your Sim can fill them rather quickly. Place a bird cage on the lot, and feed, talk, and let the bird fly free. (This helps raise your Sim's interest in pets). Place a dog-cat adoption cage on the lot with other pet paraphernalia like beds, collars -you can also sell perfume, etc, there. A latte station or two, two swings, trees, flowers, to decorate, 2 chess tables. I also place 2 couches on the lot. Sims will gather there and meet new Sims.
Fun things for Sims to do outside helps.

Perhaps your Sim's interest is low for pets. You can boolProp testingcheatsenabled true and raise your Sim's interest in pets.