A very sensible question OP...
On the face of it, yes. Dogs are like children, especially puppies.
They need toilet training, not too difficult but it takes time and patience and this varies depending on the dog.
They have to be walked twice a day and bathed and brushed or they smell and moult all over the place.
You need to spend lots of time with your pet, they are emotional creatures prone to stress and depression like humans.
Larger breed dogs such as labradoors, german shepards, boxers, springer spaniels ect.... are VERY energetic so they should be walked an hour in the morning and again in the evening and you cannot get away with not doing it because it's training/snowing/your ill.
Even with smaller dogs, they still need to be walked and cared for.
If you DO get a dog I suggest a small breed dog, short haired.
You also need to weigh it up, certain breeds are not so good with children, for example KCC are prone to arthritis in their back legs and can get snappy, Staffs are cute but get extremly powerful very quickly.
I have a Labradoodle called Daisy and she is fantastic with the kids, but totally bonkers and has to be walked for hours, she also eats EVERYTHING, that's the Lab in her apparently.
You need to do a lot of research to decide what breed would be good for your family.
I also think you will totally have your hands full with a pup and a toddler, even more reason to either put it off or got for a small, short haired breed.
You also need to think of the expense, you will have to pay, even for a rescue dog, usally about £100+, more for a pedegree dog. (I paid £300 for Daisy).
You also need to be smart if you get a rescue dog. A lot of rescue centers put the dogs on something called a "haltie" when they walk them out for you, this TOTALLY alters their behaviour and makes them docile and well behaved, so if you do look at a dog you need to see how it behaves off the lead/collar. The difference in the animal can be quite alarming.
A responsible owner, will get their pet vaccinated; £99 vacs for life with Vets 4 Pets, microchipped;£25 and Pet Insurance; this varies but I'm paying £24 PCM with DL, a bed, lead, collar and bowl will be another £50.
You also have to worm and flea them every 4-6 weeks.
I spent £500 when I bought Daisy.
If your dog has to go to the vets that's more expense, P.I only covers certain things and most P.I companies make you pay an excess, mine is £80.
Daisy once ate something and vomitted, I had to take her to the vets the day before pay day, I literally had no money and NONE of the vets I called, including the vets she was registered with would see her unless I could pay, I had to borrow the money off my mum and pay her back the following day.
Just some info to think about.
Good luck! x