A Good Hose
You always need water somewhere. Get a 1/2", 5/8" or 3/4" hose the really small ones will drive you nuts. 5/8" being the most popular, 3/4" may be a little heavy, especially if it is long and full of water. Be sure you get a hose that is not too stiff, you will be fighting it forever. It will kink on you and you have to go back and fix the kink, and probably get a cold bath to boot. Many good hoses have a "lifetime guarantee" so keep the original wrapper someplace in the garage or garden shed in case you have a problem, take it back where you bought it. If the company is still in business they should replace it, unless you run over it with the lawn mower or crumple the end with the car tire, that is not their fault. You would be surprised what some people expect.
A Good Trowel
Almost every gardener will need a trowel at some time. Bedding plants, bulbs, fixing a sprinkler head. But I would recommend the heavier ones not the really cheap pressed steel, something with a strong shank either steel or aluminum. The cheap ones will break off at the handle fairly easy, they may be great for a toddler if they want to help you, but if you are a serious gardener you want the stronger ones. It is cheaper in the long run.
Because some branches are too large to cut with a shear or lopper. You may break a very costly tool. Don't buy just any "hand saw'. Pay a little extra and buy the good ones with the blade made in Japan. All of the others are a waste of money and too much work. Even Fiscars who make most of their tools in Switzerland and considered to be one of the very best, use the Japanese blade. Also when you are done, spray a little WD40 on it to keep it from rusting, it is too expensive to neglect. If you have not used one of these saws (they come in many brands) you don't know what a pleasure it is to saw a limb.
Hula Hoe (or similar)
These are not hoes like the ordinary hoe. They are strictly for weeding. Some are called a "scuffle hoe" but are similar in how they work. They cut off the weeds just at or below the surface. You can work up right around plants without hurting them. Rake up the cut-off weeds so it looks "nice" or leave them to just turn brown. These work really well where you don't want to use a chemical weed control.
If you have hard soil, and who doesn't in our area, a pick will save a lot of work digging holes or making trenches for irrigation lines. You won't use it every week, but when you need it you will be glad you have it.
Plants grow. Sometimes they grow in directions you don't want them. Sometime they grow too large and need to be cut back. You may need to cut blooms or fruit. Choose one that fits your hand comfortably. Good quality shears will hold a nice sharp edged longer than an inexpensive one, and they usually take less energy to cut as well. Spray them with a little WD40 after use to keep them from rusting and they will last for years. Most have RED handles so they don't get lost in the trimmings, look for the RED handles ones, GREEN is not so good (blends in with the trimmings and may get tossed). Just little things to think about.
Shovel (round point)
Everyone needs one, how are you going to dig a hole? Oh you have a dog! But they dig in the wrong place. A shovel will always dig in the right place. A round point does it a lot easer than a flat one. Don't "cheap out". Buy a good one it will last for years. The cheap ones, the handle will break or the blade will split. It does not have to cost a fortune, but the $9.99 one won't last. A tip, if your soil is hard, dig a small hole, maybe five or six inches deep and fill it with water, twice. Then come back in two days and you should be able to dig the hole with little effort. What you don't want to do is struggle so hard you bend the shovel or break the handle. This works well 95 percent of the time. For a fifteen gallon container you may have to dig a larger hole and fill that with water, and come back again in two more days, just don't kill yourself, or the shovel.