Alrighty:

I work full time, so Monday to Friday I'm out of the house by 7am, and come home around 6pm, but on weekends I like to stay home and only go out to get groceries. Basically this means that I have weeknights and weekends to interact with the dog, is this enough? I'm worried that the dog would get bored or lonely when it is by itself.

This is going to be too much time alone for pretty much any dog. If you're out of the house for such long periods of time you'll need to be able to invest in a dog walker to come by and/or doggie daycare. Which leads me to:

My partner wants to get a puppy because he thinks that a puppy would bond with us more easily. I love the idea of getting a cute little puppy, but I read that puppies require a lot more work and attention at the beginning, especially when it comes to toilet training. Of course, there's no guarantee that a mature dog would already be toilet trained, but I get the impression that a mature dog would be easier to deal with.

Puppies are indeed hard work. I suggest you read this wonderful post by /u/lucked up, which gives you a good indication on how much work they are. I'd say with your schedule a puppy is definately a no-go and as I mentioned before, even an older dog is going to struggle being home alone 7am-6pm every weekday.

Off the top of my head, I'm thinking I would need dog food, food/water bowls, dog bed, dog house for the back yard, collar, lead. What else would I need?

Again, refer to the post I linked above. Yes, you'll need that stuff but you'll also need to pay for vaccinations, possibly desexing, grooming, daycare/walkers, etc. My shepherd recently got sick overnight and ended up needing a $6000 surgery, so I'd suggest also looking at pet insurance.

As a huge fan of Inspector Rex, I would love to get a German Shepherd, but having one available at a shelter seems unlikely. I browsed some of the dog shelter websites and most of the dogs available for adoption are crossbreeds. Given that I have a small back yard, and not as much time during the week to exercise the dog, I'm thinking I should avoid breeds that require a lot of activity.

No no no. For the love of all things holy, please do not get a German Shepherd. Unless you're an experienced dog owner, can commit to intensive training and socialisation and hours of excercise a day there is no way you should get a GSD. Honestly, every dog - even little ones - need at least a good walk every day. Not even a big yard is going to be enough to adequately exercise most dogs.

Should I be spending a lot of time with the dog so that they get used to my presence? Or should I leave it to get used to its new surroundings?

You need to spend time with them, for the most part. If you get a puppy you may actually need to leash him/her to you when they're not asleep. Even adult dogs need training and socialisation

I know there are a lot of websites out there with information about this, and progress depends on both my effort and the individual dog, but it would be nice to hear other people's experiences. My biggest concern is toilet training. My mum's friend's dogs were the most obedient and well behaved dogs I ever met. They never peed/pooped inside the house, and in the back yard they only did their business on the lawn, never on the pavement.

The easiest way to toilet train is crate training, but you need to do the research on how to properly do this. It can be time intensive and you need to do it properly - proper introduction, behaviour in crate, knowing how often to take them out to poop/pee and how to reward them when they do, etc.

Honestly - and I mean this with no disrespect - you need to do a lot of research before you even think about getting a dog. My opinion is - with your schedule alone - now is probably not a good time. If you do decide to go ahead read pretty much every thread that's even slightly relevant on this sub, /r/puppy101 and /r/dogtraining . Also, fill out the breed survey on the sidebar. But really, if you can't put in the time and effort to find the answers for these questions yourself I don't think you're ready for a dog