Today is my first day at a new role. I am very optimistic about this role.
At the same time, some of the places that I applied are only now following up. One role in particular has the potential for a 100k+ TC increase, or over 30%. This is meaningful to me. What mental checks should I do before accepting the interview?
If I interview and receive an offer, would it be best to accept it and leave or bring this up to my current employer and seek a raise? Would that set a sour tone?
If you're asking your current manager/company for a compensation increase, you should have data to suggest that you're a strong performer, i.e. wait for at least one performance cycle.
Gaining leverage through a better offer is much more effective when you haven't accepted any offer. Trying to get a raise a month after joining is not a good look and you may strain the relationship. I would not recommend it unless you feel like you were shortchanged, or the other offer is substantially more.
I talk about this in the negotiation course: Negotiating Your Current Compensation.
If the new offer is a 30%+ TC increase and has more potential for growth, I'd still consider it. Be clear about (1) the interview process, (2) the timeline they're looking for, and (3) what is important to you.
Basically, you want to have a framework you can articulate to yourself or other companies about why this is an exceptional case worth interviewing for.
I recommend following the advice here:
If I interview and receive an offer, would it be best to accept it and leave or bring this up to my current employer and seek a raise?
You definitely shouldn't seek a raise. A newbie asking for a raise would be pretty crazy (and has near 0% chance of success). Just take the other offer and hope they don't rescind it.
Zooming out, I would be careful about interviewing on the side at a new job, especially one that you like. Onboarding is very mentally intensive, and you don't want to mess that up by splitting your mind space between your current job and other opportunities. I recommend only following up with these other roles if you're in the later stages (ideally onsite), and you're confident it's a better role than your current one.
Check out the onboarding course as well: The Complete Onboarding Guide For Software Engineers