You have come to the right spot for advice!
I started brewing recently as well. Like you, I started brewing with my dad when I was 11-16 or so. It was a great experience with pops.
Anyways, that was 15 years ago and homebrewing has come a very, very long way. I received advice from pops a couple months ago when I mentioned that I was going to start homebrewing and the one thing he said was to use sanitize with warm water and colorox and rinse everything extremely well. Boy am I glad I came here first....learned about a little thing called Star san..."don't fear the foam!"...I digress.
First off, I purchased northern brewers beer kit for like $89 off amazon. Much of that expense was due to the glass carboy..which consequently I don't even use because I don't rack to a secondary vessel. Those things are a P-I-T-A to clean as well. But it gave me everything else I needed for bottling and racking and fermenting so...not a bad deal. You could piece it out if you wanted cheaper. The food grade buckets are only about $14 at my LHBS and can be had cheaper online. Here is a list of everything I use on a brew day equipment wise:
1.) 8 Gallon pot - I would recommend you get a 10 gallon pot at least. If money is tight, then a 5 gallon at walmart for $20 will get you started on extract. BIAB will be difficult unless you plan on 1-3 gallon batches.
2.) Stainless steel spoon - You can buy a paddle if you want. I prefer the spoon. To each his own.
3.) Hop bag - Get one from BSG or something. It'll help keep your beer a little cleaner.
4.) STAR SAN - get the big bottle from Amazon. 26 oz. I think it's $26 or something. But it concentrates to 1 oz per 5 gallons of water.
5.) Rubber gloves - Star San is not great for your hands. Any time I sanitize my bucket I always wear gloves. I dried my hands out something awful the first time. Just dipping your hand in for small parts isn't a problem. Just rinse your hands real quick after.
6.) Digital Thermometer - just a cheap one will do. Temp is very, very important. Especially if you go to all grain BIAB.
7.) 20 pound bag of ice for the ice bath. It takes a lot of ice. Unless you have a wort chiller.
8.) Fermentation bucket - Please get one with a spout near the bottom. You can avoid a siphon if you get one with a spout...for the most part. Depends on your trub bed.
9.) Air lock
10.) 6 foot hose for transferring
11.) Hydrometer
12.) Thief
13.) Gallons of Spring Water
14.) Bottle caps
15.) Bottle capper
I think those are all that I use on brew day. I don't use the hose for transferring until keg day, but you'll still need it at some point.
I did 1 extract kit to start out. I did a brewers best milk stout (stouts are pretty forgiving). The only downside is the take longer to age for the anxious beginner homebrewer. I then found a recipe for and Irish red (not in a kit) that I had my LHBS help me fill the grain bill and the extract. Then I found BIAB, I then bought the 8 gallon pot from a 5 gallon and started all grain. Haven't looked back. All I needed to switch from extract to BIAB was a Wilser bag and a bigger pot. Easy.
My LHBS gave the following advice on getting started. Start extract from a kit. Follow the directions and learn the process. Then select a beer that you want to be your house beer and get it to where you want it and can consistently make it taste the same batch over batch. Then branch out.
Most of all, have fun! It's a great hobby with great benefits. You get to share your beer that YOU brewed with friends and family. Also, it's a money saver, which for me, is always good. I am an accountant and have broken down the numbers. I am definitely saving money doing something completely awesome.