Note: some of the below links are affiliate links, however, they're products I personally use or believe will be of help to you and cost you absolutely no more. I make a small commission on some of the products if you purchase through the below link.
Seed Starting & Containers
Row Covers
Seed Organizer
This is a plastic photo organizer that perfectly holds seed packets and allows you to keep them extremely organized.
Seedling Heat Mat
If you're home is cold, you might consider using one of these for your warm-weather starts. I've not used it, but it comes highly rated and recommended.
Large Grow Bags
These fabric grow bags come in multiple sizes and are an excellent way to grow vegetables.
20 pack of 1 gallon pots
I purchased these three years ago and they've worked well with repeated uses. I put my tomatoes and peppers in these when they reach about 2 to 3 weeks of age.
20 pack of 4 inch pots
These are great for smaller starts that you'll be putting out sooner, such as cabbage, broccoli, or even herbs and flowers.
Frost Protection Plant Fabric
For extending your growing season and protecting plants from both frost and insects.
Container Potting Soil
(Costco has the best deal I've found in the springtime) but I've also ordered this brand in the past
Newspaper Pot Maker
Just use newspaper and this roller to make all your little seed starting pots
Greenstalk Vertical Stacking Container
Here's the vertical stacking and rolling system I have for my strawberries. The stackable vertical container garden tower is from my affiliate GreenStalk use this coupon code for $10 off your order: PIONEERING
Grow Light Options
Grow light on a clamp
This is a smaller grow light for small batches of seeds.
Growlight System with Pulley Rack
This is the grow light I've had for over three years. It folds up compactly and I store it under our bed. Still using the original light bulb as well.
Tomato Success Resources
Heavy Duty Wire Cage
Supports for tomatoes, I have used both the smaller and larger ones for multiple years. My tomatoes grow so well, no matter which size I use, by the time they're full grown and producing tomatoes, I still use some twine and rope to finish off their support.
Gardening Tools
Shovel Handle Attachment
Back Saving Gardening Tool Universal Fit Compatible with Snow Shovel, Rake, Spade, Weed Eater and More
Watering
Soaker Hose Kit
Works great in many areas of your garden. Easy to install and maintain.
Soil Testing Supplies
Simply Soil Testing
Using a lab will give you the most accurate results. Check with your local county agricultural extension office to see if they provide this service, most do (mine being one of the few states that don't). This is the lab I use as my local county office doesn't test soil, the basic test is only $12 and you can do it all by mail!
Luster Leaf Rapitest Soil Kit
For an at home test, this Luster Leaf Rapitest Soil Kit is rated as a best seller.
After looking at the reviews on probe soil test kits, I didn't find one that I could recommend to you as reliable.
Gardening Books
The New Vegetable Growers Handbook
by Frank Tozer. Excellent resource on growing vegetables, when to plant, how much, including seed saving (though not as heavy on seed saving as Seed to Seed). This is my go to reference for growing vegetables and troubleshooting problems.