I'm just getting going on hand-me-down bikes myself and have started putting on some miles so I've been kinda learning on the fly like you. Glad I'm not the only one who got started on yard sale equipment. LOL
The easy stuff is... first, make sure the seat and handlebar height are right...or as close as you can get it. Nothing makes your knees hurt like a too-low seat. Bars that make you reach down so you feel like you're standing on your head will soon wear out your neck and shoulders and the bike'll be sitting in the shed. Get a decent seat or a gel-pad until you get callused up or wear bike shorts with pads. Your cheeks will thank you.
You've already got it tuned up so just put on a bottle rack for liquid if it doesn't have one and go. A tool kit ain't a bad idea if you're planning on getting too far away from home. Some basic stuff with you, a wrench or two, tire patch, chain kit etc. can mean getting back or calling for a rescue. I should know, my friend broke his chain today and guess where we both had our repair kits...in the bags on other bikes. I should take my own advice.
From there on out...it's up to you and your wallet how much more you want to do. Older bikes may not be worth the effort to upgrade too much but so what? I put an awful lot of miles and had a blast on a bone-stock mid-70's (I'm guessing) Raleigh Record without doing anything more than you've already done to your Schwinn and I'm still riding a 15 year-old Trek that's almost out of the box and love the thing.
Have fun
I feel that foam grips vs. bar tape can be a bit of personal preference, that being said, I think that bar tape is much better (cork is awesome, synthetic cork lasts longer).
One of the best things that you can do to improve performance on any bike is to adjust and fit everything to your body size. I found out last week just how much of a difference 1 cm increase in saddle hieght can make. The best part about doing this is that it's free.
Agree on the pedals suggestion, that's definitely where to start. In terms of personal performance, I found getting a fairly basic bike computer was a real plus. Keeping a track of how far, how fast and for how long motivates me to ride harder regardless of what I do to the bike.
if the one you have is a good, light-ish frame that fits you well and feels nice to ride you can upgrade as much as you like (afford). Ultimately if it's a cheap/heavy frame you will end up replacing the whole bike and in the short term have wasted your money on good parts for a cheap bike.
agree on the pedals and computer.
buy some quality chain lube and use it.
Keep it tuned well, buy some better tyres (lighter perhaps), make sure the wheels are true and ride it until you can afford a better one.
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