Is it better to invest funds, or dedicate everything to paying off student loans?

I am 2 years out from grad school and officially entering the stage in my career where I can attain my certifications/licenses. I've also recently signed a contract to transfer to a clinic closer to home, bumping my pay from $20.19/hour with 40 hour weeks (technically $42k annual gross, but last year I made about $55k net with OT and bonuses) to $60k salary with an additional $5k per certification as I complete my board exams in each discipline within my field.

From grad school, I had roughly between 70-80k in loan debt, I don't remember the exact amount, and over the last 2 years I've paid it down to $58,500ish. I'll provide more of the specific details below in a bulleted list.

My strategy so far has just been to remain on an income-driven repayment plan, and as my savings goes up, I occasionally make lump sum payments into my accounts — generally when my savings hits 30k I've dumped 10k into the loans.

My question is basically if it would be better to keep a lot of that savings invested instead. I know next to nothing about investing (read: I know nothing), but my brother let me know their are taxes when withdrawing from investment accounts. It just feels silly to keep money in a savings account as a 'what if' and not let it accrue better interest.

The details of my accounts are as follows:

Student Loans:

• 1 Consolidated Loan at 6.375%
  • $58,576.28
  • Accrues $10.15 in interest daily
  • Current monthly repayment is $49.63
    • I am definitely considering changing plans to adjustment my monthly repayment to cover all interest at least
• I owe my parents roughly $20k for loans they provided me for housing while I was in school, there is no interest that they care about, so that all is waiting until after the government loans are handled.

Bank:

• Checking: $100
• Savings: $507.58
  • 7.01% interest based on average daily account value
    • Last year accrued basically $2
    • Interest rate decreases past $500
• Money Market: $20,212.57
  • 2.58% interest
    • $40ish last year

I only have a Discover card which I use for essentially all purchases that do not have larger fees and I pay off the balance monthly.

So, what do you all think? Better to invest? Better to continue as is, or devote more of my savings/income to repayment? I don't intend to change my lifestyle with the raise, ideally I want to pay off all the loans in the couple years before I'm 30, but for the most part my strategy when it comes to money is 'just don't spend it.'

Anyways, let me know what you think. Thanks