Looking at the vastly uncharted domain of applying to universities and talking to professors or to Industry professionals for Internships does get nerve wracking. As someone who has been through this, I agree it is not an easy task. But with the right approach, it is definitely possible.
DISCLAIMER : These tips are based on my experience only, and from what I have seen amongst my peers and collegues, generally tends to work well for Tech and Design domains. But the latter tips may not be so effective for roles such as Finance or Consulting.
The first step is to build an Online presence. The most common and useful app is LinkedIn. Create a profile, add a professional picture, have a crisp Bio along with the relevent experiences and skills added to the appropriate sections.
Once you have a decent profile set up (All the above things), start building up connections. The way I did it was to send connection requests to any and all students from my university, and to send a thoughtful connection request with a "Personalized Invite" to all the professors/professionals of interest to me. During this time, engage with people whenever you can, on posts you find interesting (maybe commenting on them, or messaging people directly to just talk about their work/research). It will help in complimenting your online presence.
Once you have around 500 connections, you are good to go.
Ideal Time to do this : Yesterday. DO THIS ASAP.
You should start creating a Database of companies and professors you find relevant to your field. There's no need to give a lot of time daily to this. Around 30 mins to google and look up companies/professors doing research/work interesting to you would suffice. This database will be very handy when you start sending emails, and for keeping track of whom you applied to.
Useful ways to look for companies :
Ideally, if you are looking for a company internship, start around August, and apply/send mail to a minimum of 3-5 recruiters/companies daily. Be bold! I've even secured an interview when I directly texted the CEO of a company on LinkedIn. Also, remember to keep track of who you mail/apply to.
Useful ways to look for professors :
Ideally, if you are looking for university research internships, you should begin around July-August, and look for a minimum of 3-5 professors daily, and send them a mail. Add them to your database and await a reply. Make this a daily ritual, to look for professors, mailing them (It usually takes an hour or so). Alternatively, you can select one day and do a whole week worth of mailing on a single day. Do remember to schedule mails in either case, so that you can increase the chances of the professor actually reading your mail (more on this later).
Put yourself in the shoes of a person at a company trying to hire somebody for a role, or a professor looking or relevent interns. Many people apply through the usual channels - LinkedIn or Email. While this might get noticed and selected, there's definitely a better way that can you use to increase your chances. You have to try to get the recruiter/professor to notice you or your profile by choosing an unorthodox way. And the best way to do so is using Information sharing platforms (Other than the most common ones used for recruiting like LinkedIn). They can also be used to directly search for potential opportunies in the databasing step itself!
Go on your usual information sharing platform and look for interesting work. For example, if you are into Web Development, you can go on Github and look for interesting web designs/implementations, and directly talk to those people. If you're into design, you can check our various portfolios on Instagram or Behance, and talk to those people directly. People LOVE when others show genuine interest in their work. Hardly anyone would turn you away.
Some information sharing platforms:
This way, you arent competing with hundreds of applicants for their attention, but instantly catching it by doing something out of the box. Usually, the recruiters/professors are looking for talent with working experience, which can easily be portrayed by your discussions/contributions on the Information Sharing platforms.
If it is a company, have patience, and let them reply. Usually, they will always revert back to you, and not leave you hanging. Don't do the same to them. If you have a better opportunity, and want to pass this one up, tell them upfront and avoid wasting your and their time. Deferring to reply for a few days to think it over is fine, but ghosting them for waiting to hear back from another company isn't.
If it is a professor, keep this in mind - think of the time you are sending the email. Sending it on a weekday during the day will have a higher chance of him/her reading it (Preferably Wednesday and Thursday). Sending it on the weekend might mean they ignore it then, and have to look at it along with the huge chunk of mails that have piled up over the weekend, reducing your chances. Always better to be safe rather than sorry. Schedule your emails accordingly, and keep an email tracker (This one for gmail works nicely. Another Alternative)
Give them time to reply. Don't send a reminder mail unless it has been 8-10 days. IIT students are notorious for persistently bugging them. Don't give them a reason to not select you.
I mainly started in around June itself (people would say its wayy too early, but I dont). The first 2-3 weeks were spend in databasing companies mainly, and extending my Linkedin Connections. It only took around 30 mins daily to add new companies to my google sheet. Sending Linkedin Connection requests would take around 15 mins - the same time it would take for morning digestive business. Hence, that was the perfect time lol. This went on for around a month. After this, I started sending emails and applying for positions. In hindsight, I should've waited for another month or so, because that was too early for companies to begin recruiting.
Another mistake I made was not being open to the idea of a university research internship, which is why I din't apply to the Mitacs Globalink Program - a huge mistake. Keep your options open. Because a university research internship is where I eventually ended up at. Other than that, over the course of the next few months, I applied and gave interviews at a lot of companies. And was rejected at almost all of them. For a variety of reasons - sometimes it was the coding round, sometimes it was that they needed offline interns, which wasn't possible at the time. One thing to take away from this is - PATIENCE. Be Patient. Please. You WILL face rejections. For sure. Loads of them. I faced more than 100 of them. But eventually, you will DEFINITELY end up at a place that's really good for you. In hindsight, I'm really grateful I din't give in to the FOMO I was facing in the initial months, and hence did not for roles that I dint wanna do. I am pursuing work I really like, and that took time to find. It was all worth it. It always will be.
Internship Season is daunting when you see that students all around you are being shortlisted and selected, while you are merely lengthening your database. But patience pays. Trust me, it does. Instead of being stuck with an intern that only half interests you, wait and spend time finding one you throughly love working at.
Also, don't ever be afraid to apply for a role/mailing a person because you feel "Incapable" or "Inadequate",or are just downright scared to do so. I literally secured my internship because the professor I'm currently interning for, put up a post on LinkedIn asking for PhD candidates, to which I replied, "I'm an undergraduate, and I'm really interested in your work. Any chance of a summer internship?". And then things progressed from there. As they say, "Poochne me kya jaata hai" (What's the harm in asking?)
If you have any questions, don't hesitate to reach out!
Keep going!