Hey and welcome to my guide!
I've compiled just about everything from levels to achievements and how to get those rather easily. This guide is for people who just can't seem to get past that one level or people who just want to see everything this game has to offer!
I really like this game and picross in general and I often see people struggling with these kind of puzzles, which led me to making this guide.
Now let's get to it.

This will unlock the 'A Good Boy's World' - Achivement

This will unlock the 'What's Up Dog?' - Achievement

This will unlock the 'Pug Life' - Achievement

This will unlock the 'Hot Dog' - Achievement

This will unlock the 'Spot On!' - Achievement

This will unlock the 'Branch Manager' - Achievement

This will unlock the 'Ruff Around The Edges' - Achievement

This will unlock the 'Bad to the Bone' - Achievement

This will unlock the 'Ground Dog Day' - Achievement

This will unlock the 'Now Do You Believe in Dog?' - Achievement

This will unlock the 'Cat is out of the Bag' - Achievement

This will unlock the 'Cool Cat' - Achievement

This will unlock the 'Prince of Persian' - Achievement

This will unlock the 'Feline Good Meow' - Achievement

This will unlock the 'Thick Skinned' - Achievement

This will unlock the 'Pawsitive Attitude' - Achievement

This will unlock the 'Abyssi Nyanyanyanyanyanyanya!' - Achievement

This will unlock the 'Purrfection' - Achievement

This will unlock the 'Nine Lives Aren't Enough' - Achievement

This will unlock the 'Eye of the Tiger' - Achievement

A lot of achievement are obtained by doing specific things in the garden. By now you should have unlocked everything (Which means you also got the 'Every Doggy' - Achievement and 'Puppy Cross Master' - Achievement), so there shouldn't be a problem with completing these.

Put one dog and one kennel-type item in the garden together.

Put 10 items into your garden.

Put two Italian Greyhounds, one Greyhound and one Chihuahua into your garden.

Put each type of toy and a Bull Terrier into your garden.

Put the bird kennel and a Beagle in your garden.

Put one cocker and one schnauzer together.

Put one Scottish and one of each kennel in your garden.

Put three bones and one Westie in your garden.

Put a Greyhound and a donut together.

Put one Shih-Tzu, one tree, an insect, one flower, one kennel and one watering can into your garden.

Put one Basset and one duck pond in your garden.

Put at least five Corgies in the garden.

Simply pet your dogs 100 times by right-clicking on them. You can actually pet them faster if you manage to get the sprites to overlap, so that you're petting multiple at once.

Put one of each dog into your garden.

Now, I personally recommend that you should find out how to access the dalmation mode on your own, but if someone is stuck and doesn't know where to turn to: Have you tried clicking on the Dalmation on the main menu screen? You can also click on the Shiba-Inu to see your achievements

You will have to complete at least 102 picross grids to get those achievements.
The 'Quantum Mastery' - Achievement requires you to play a 20x20 grid, but you can just complete 5x5 grids for the 'Call to Explore', 'Randomizer' and 'One Hundred and One Dalmatians' - Achievements.

I also recommend playing the credit levels on your own to honor the developers, but if you can't solve them and still want to see what they look like:
here they are completed.