Getting the offer is the start. What you do in the first few months is what turns an internship into a reference, a network, and sometimes a full-time offer. This is the advice I give students once they have one lined up. For getting there, see Interviewing for internships.
The first few weeks
Nobody expects output yet. The first few weeks are for getting your accounts and access working, reading whatever documentation exists, shadowing people, and learning names. Use them to figure out who knows what and what the team is actually measured on. The pressure you feel to prove yourself on day one is coming from you, not from them.
Learn from everyone around you
Ask questions, but do a little homework first. You are surrounded by people who know a lot and are usually happy to share it. Try to find the answer yourself, and if it takes more than about 30 minutes, ask someone. Your manager would much rather answer a question than watch you lose a day to it. No question is too basic, and if something is undocumented, others probably have the same question.
Document what you learn. I write down almost everything: meeting notes, project tracking, and notes on technologies I am learning. It cements the knowledge and creates documentation that did not exist. Keep a separate running list of what you shipped and what it changed. You will need it for the resume update and for the return-offer conversation.
Work with your manager and your team
Tell your manager your goals and ask for feedback. They are the person best positioned to help you navigate the company and get where you want to go, and often the person who can convert you to full-time.
Make your manager’s life easier. If a tool or process is causing them headaches, try to find a fix.
Be honest about your skill set. If you are asked to do something you have not done, say so, and say it may take you longer.
Reach toward the work that interests you. I was on a security operations center (SOC) team and curious about reverse engineering, so I asked that team whether they had something I could help with. I kept my manager in the loop and did not fall behind on my main work. That is how I moved from the SOC into reverse engineering.
Be kind, respectful, and calm. Get to know your coworkers as people, and show up to company events. If someone is hard to work with, take a break, reset, and ask them directly what they need from you. If that does not fix it, tell your manager.
Grow deliberately
Get outside your comfort zone. If you love one kind of work, deliberately take on other kinds too. If you write detections, go learn how the logs get there. If you code, take a shift triaging alerts. I have been a SOC analyst, a reverse engineer, a software engineer, and a security engineer. The more of the system you have touched, the more valuable you are.
Be compassionate with yourself about imposter syndrome. The further you get, the more aware you become of what you do not know. That feeling is normal and does not mean you do not belong.
At the end
Ask about coming back. If you want a return offer or a full-time role, say so directly and ask when that decision gets made. Managers cannot advocate for something they do not know you want.
Ask for the reference while the work is fresh. A short note from your manager, or a LinkedIn recommendation, is much easier to get in your last week than six months later.
Keep in touch. Two or three people you clicked with is enough. That is the network, and it is how the next opportunity usually shows up.