Assistance with luggage at Canberra airport for a woman travelling with children

This answer was thought up before the age of the children was added to the available information, it will be easier with children a bit older than the one and three year old ones of the OP.

While someone to help with the children will be appreciated, it is not needed. (And if there is a helper, I would want someone on and just after the flight, not just at the time you pick up your luggage.)

Airports are designed so that people can get through and out with the least difficulties. This means that for standard travelers no extra help is needed.
Traveling with young children is not standard but you can do a lot to make it less of a problem.

I expect a three year old child will be able to walk fair distances, a one year old child will not be able to walk that far (if at all.)

Travel with a sensible amount of luggage, (other answers and comments give ways to restrict the amount,) like a cabin size rolling case or smaller rolling bag for hand luggage, a small rolling case or small backpack for the bigger child so they can feel big, (but never put important items like their favorite toy in it,) and as little else as possible.
And no more than three pieces of check-in luggage, each of them no heavier than you can handle by yourself.
If you can gate check the pram/buggy, try if they will hand it back to you at your arrival gate. If not, see if there are any local replacements or small luggage carts available for arrivals.

If you can not use a buggy or some such, carry the younger child, (child carrier or even sitting on your hip supported by a long scarf tied the right way,) and have the bigger one walk alongside you. Best use a child harness or backpack harness with a leash for the child so you do not have to force it to hold hands for a long time.

Getting to the immigration area will not be much harder with two children, although they may get tired and less cooperative when they day gets long. Getting them a drink or small new toy may work.

When you get to the ‘collect your luggage’ step, find a place to sit down away from the carousel, keep the children as occupied as you can, while you wait for your cases to be available. Before you go to get the cases, make sure you have a trolley. (In some airports you need exact change in the local currency or a credit card.)
When your luggage is on the carousel/band, you tell the older child he is to guard the carry-on and/or the smaller sibling. Tie his leash to the carry-on or buggy, so he can not follow you.
Step forward at the best time to get your luggage and put one on your cases on the trolley. (If you carry the younger child, it is not unlikely that someone will offer to help you, if not and you feel you need help, look around and ask someone.) Just take the one case and go back to the kid(s) and go out again when the next case is ready to come of the band. (If there are helpful people, you may want to keep both children with you, just far enough from the carousel that they can not reach it.)

With the luggage on the trolley, you walk through customs, where they may ask your questions and need to open your luggage, or you may want to show them items you brought which may need special permission. (All food, wood and so on.)
As soon as you are out of that you will be in the open area of the airport where your husband can wait for you.

This system will only work if your older child (or both) are used to the harness (or backpack) on a leash and are willing to do what mommies tells them. Being in an airport is not the place to discuss and convince your child, when you tell them to do or not to do something, they should do or not do as you tell them.
And the older one should be used to guarding the sibling or taking ‘care’ of the luggage as told. An almost 4 year old should be able to, a child just turned 3 will find that a lot harder. Children who are a bit older will be much better when they feel this is when it is important.

Expect one or both of the children to cry or have a tantrum while in the airport. Ignore that as well as you can (after you have made certain it is not something you can change) as children will make noise, especially at the end of a long day and everybody else in the airport has experienced it themselves.
If the people around you do not like it, they can take themselves away, children cry, that is natural.

Do not expect strangers to help you with your luggage or children, but when they do accept it (to a point, you do not want them to carry your child or your important items where they can easily leave you.)

Since it’s your and your wife’s return flight being separate, there are multiple options to reduce the direct physical load on your wife:

• You can take most of the luggage with you, when you depart. For example you can pack all laundry that has been worn and only leave your wife with as much as needed for the time left. Probably you’ll have to check in extra/overweight luggage.
• Your wife could simply not claim the luggage at Canberra. Unclaimed luggage will probably be moved to a storage area. You can try claiming this later, e.g. the next day. You should try to find out, what Canberra airport does with unclaimed luggage. Most airports auction unclaimed luggage after several months. It’s possible that they charge a fee for extra handling in case of late claims. Still, this is probably cheaper than a domestic flight. It comes at a slightly increased risk of losing your wife’s luggage due to the extra handling.
• If there’s luggage that’s needed after you left and before claiming “unclaimed” luggage is possible, your wife can get a trolley before claiming it. This is also a solution for luggage, which you don’t want to expose to the slightly increased risk of losing.
• Your wife could send part of the luggage home via parcel service. International shipping from Doha to Canberra is most likely expensive although not as expensive as a domestic flight just to meet at the airport. This has the advantage over you taking stuff with you, that your wife could send all the stuff (except for the most basic) on the last day of her stay. Make sure not to send brand new electronics or stuff that customs might take an interest with. If you decide to send electronics, make sure you still have the bill at home.

Combining 2-3 of the above options should account for loads of stuff. My brother attempted another option in the past:

• Take single use stuff and throw it away: He bought a collapsible, very lightweight used trolley for his little girl to ride in and intended to throw it away after the vacation it was bought for.

In the end he tried to stuff it into a dust bin at an airport, because on the return flight it suddenly counted as extra luggage (which it has never counted as before). This was three vacations later than originally intended, because it turned out to be more sturdy and handy than expected. Btw. it didn’t fit the dust bin and he was allowed to take it with him without paying extra. However the principle can work out, if you can let go of stuff.

International and Domestic flights do arrive into the same terminal in Canberra, on the grounds that there is only a single terminal.

However, the international arrivals area is completely separated from the domestic arrivals area. If you were arriving on a domestic flight then the first opportunity you would have to meet them would be after they have already been through immigration, baggage collection and customs – at which point they would be let out into the domestic baggage collection areas which is a public area of the airport that can be accessed by anyone (even if not arriving on a flight).

As a domestic passenger, you would have no way to access any of immigration/passport control, international baggage collection, or customs.