Like others have said, tasting a variety to find what you like is key. Here's some recommendations of excellent bottles that are easy to find, won't break the bank, and will give you a taste of the major brands/flavor profiles:
**These are the prices I regularly see outside of Austin.
1. Russels Reserve 10 Year - $40 - High age statement, incredible value, one of the best from wild turkey that's always on the shelf.
2. Knob Creek Single Barrel Select 9 Year - $60 - High age statement, higher proof at 120, great flavor profile that will change your mind on Beam products. It is a single barrel but I've never had a bad bottle and some are exceptional.
3. Four Roses Single Barrel - $40 - People's opinion of Four Roses varies wildly, but it's one of the big brands and worth exploring. The single barrel is affordable, easy to find, and is a much better quality than their cheaper offerings.
4. Buffalo Trace - $26 - This is very easy to find here in the hill country, but may be a bit harder to find in other places. BT is one of the best value bourbons out there, and most of their other bottles are highly sought after and hard to find. Most people prefer Eagle Rare 10 Year but that's still harder to find depending on where you live. It a regular BT is a great way to see if you like the profile before you hunt the rare stuff.
5. Maker's Mark - I'll let someone else chime in on the best Maker's to grab because I've never been a fan. With that said, it's very popular and it's very easy to find unique single barrel picks at good prices. Grabbing a Maker's is essential in my opinion because it's a Wheated Bourbon and a different flavor profile from the rest.
6. Elijah Craig Small Batch - $35 - This is the traditional bourbon profile, full of flavor and easy to drink. This is the bottle I always point new bourbon drinkers to because it's so affordable and approachable.
7. Old Forrester 1920 - $60 - Old Forrester has a lot of great stuff you can always find on the shelf, and their high end stuff is excellent. 1920 is the most popular OF and worth picking up.
Bonus - Grab something local. It's a lot of fun to try bourbon from your state and take a tour. You'll learn a lot and might find some gems. I got into bourbon after volunteer bottling for Garrison Brothers when they were first starting. I love Garrison and the people there, but if you're just starting out and want a Texas bourbon I would try the more affordable Still Austin. It's $35 for their flagship bourbon and they do 4-6 special releases a year that are always great. The Fall release of their new Bottled In Bond series is absolutely fantastic.
I hope this list helps you land on some of the better offerings from the big distilleries. Curious to hear what y'all would swap from each brand, if any.