Tl;dr if budget is a hard obstacle you could use the tub as a kratky tub for lettuce (no pump needed) to save some money and then decide what to upgrade after your first harvest.
Fun stuff! I've been kooking around myself and have some thoughts below by item, but feel free to take what works for you and leave the rest. Doing your own thing is half the fun anyway!
Tub: Looks good! From others' posts in here and elsewhere online you might want to think about extra light protection for the top. Some people experience some light leakage through the yellow top so you can either spray paint it black (on the inside), or cover it with something that blocks the light (I use aluminum foil on many of my smaller containers).
Pump: As u/fisch09 mentioned you'll need some additional hardware if you want to use a water pump. You will also need individual drip lines to each plant at a minimum but could potentially set up a spray nozzle (aeroponics). You can also do DWC by swapping out the water pump for an air pump, but would still need additional hardware (tubing and an air stone).
Cups: Those are fine, but you can go cheaper if you'd like. Do you eat Yogurt? You can cut up those used (and cleaned) containers into DIY netcups. Look around your house for stuff that looks like it could fit a plant - if it was originally a package for food, it's likely safe to grow in.
Medium: Rockwool works, but isn't reusable so you'll need to buy more eventually if you want to keep your own system going in addition to the Aerogarden. Sustainability and sourcing come into play for many people. If you're open to getting weird with things and looking for cheap, go to petco for some "hydroballs" instead of buying a big-ass bag of hydroton. Rinse well before use, but my lettuces are just as happy in hydroballs as they are in coir. Honorable mention is aquarium gravel (two aisles over from the hydroballs at my Petco) - that seems to be going ok as well. Rinse well before use.
Nutes: Too much to talk about here and I'm still learning this myself. I am currently sourcing these materials (Masterblend tomato formula, Calcium Nitrate, and Epsom Salt) for my upcoming kratky experiments as many folks online seem to prefer this blend rather than what I'm currently using (FoxFarm Grow Big).
Note different plants will have different nutritional needs so make sure that everything you're putting in the tub is at least similar (e.g. if you put tomatoes and lettuce in the same tub it's going to be hard to give them both ideal nutrients). Also note that you're going to want to be able to measure the pH of your tub so the plants can actually take their nutrients up.
Seeds: Yep, they're seeds! Should be good to grow :)
Light: You will see many opinions but in my personal experience I've learned the most here from trial and error. It looks like the Aerogarden Harvest uses a 20W grow light, while the one you linked states 25W. While that sounds equivalent, check the tub dimensions against your Aerogarden - in order to get the equivalent coverage you might need more than one light. One thought here is that you may be able to place your tub on a deck/porch if you have an outside space that gets the right light!
Last thought: if you just want to get going now you could skip the pump altogether and grow kratky lettuce and, after your first harvest, make changes/decide where and what you want to upgrade (if you want to keep your own system).
Any air stone would be good for a bucket as long as it can fit in the bottom of the bucket. Usually, they will state how many gallons they will work for, There are round "puck" stones that work well for me.
Masterblend Combo has served me well for nutrients.
3-inch net pots will cover most plants you want to grow (in my experience 2 inch works fine as well).
I would consider if you are safely able to make it to a Home Improvement store(in my area they offer free delivery) you can find the 3 buckets for about half that price and also pick up the support for the tomatoes as well.
For those white food grade buckets consider painting the lid and sides to block the light (this will keep your roots happy).
I went dry powder nutrient. Way cheaper! This is what I use
https://www.amazon.com/MASTERBLEND-4-18-38-Complete-Combo-Fertilizer/dp/B072F2BL9D/
1L of aerogarden liquid nutrient can do 100 feedings for $25 bucks, but for the price u can get 2.5lbs of powder nutes that can feed 300-400 times. Even cheaper when you buy larger bulk.
You either mix what you need or make a small concentrated bottles to quickly mix with water like the aerogarden liquid nutes.
Alternatively, to prevent mold etc inside your open bottle of nutes just put it in the fridge.
Right now using gh gro. Switching to https://smile.amazon.com/dp/B072F2BL9D/ref=cm_sw_r_cp_apa_fabt1_waxSFbMC0CGGV in about a month
I purchased Masterblend for my basil and it worked well. I decided to purchase it after seeing a lot of people using it on youtube. They sell a kit on amazon that comes with three different ingredients that you need to measure out and mix together (you will need a digital scale). So far I have grown Basil, thyme and rosemary using it and have had great results (better than the liquid hydro mix I bought).
I purchased the 2.5 pound kit off amazon for ~$22. I'm not sure how many batches this kit makes, but I have made several gallons and still have a bunch left: MASTERBLEND 4-18-38 Kit
The kit will come with instructions on how to mix it properly. I usually make a gallon at a time and mix in: 2 grams of Calcium Nitrate, 2 grams of MasterBlend, 1 gram of Epsom Salt.
A Masterblend three part kit, available on Amazon, will be one of the best deals. Each part is a dry component that has to be mixed in water and then added to your reservoir in the correct sequence to blend in properly. Fortunately there are dozens of Youtub videos showing how to do this.
Today, the smallest kit is
4.74.7 out of 5 stars (3,798) $24.99 $24.99