I love to hear team leads wanting the people they are supporting to grow. Investing in singular individuals will make the team stronger, but I'm feeling that you are wanting to nurture a overall team rapport.
Some ideas in no particular order:
1) Organize a social outing after hours. Getting the team together away from work is a good way to build rapport. You can not control interest and attendance, but if most of the team turns up you'll see results.
2) Order lunch in for everyone and take an hour in a meeting room. Sharing a meal is a good icebreaker and allows the team to socialize away from their desks and workload. You can control the attendance here. Who doesn't like a free lunch? I've always enjoyed learning about coworkers over pizza.
3A) If possible, shuffle the pairs of individuals who are working together. An added benefit to this is the strengths and weaknesses of the team/pairs will bubble to the surface and you can begin to tailor improvement & reward strategies.
3B) If you cannot shuffle the pairs, Is there another way you can get them working together? Smaller side projects, initiatives? These could fall under group objectives also.
4) Have them brainstorm the team values/mantra/name/logo/team colours. This can be done via google sheet and then discussed over a short meeting. Having all of them agree on what the team represents will increase team pride and provide a banner to unite under. For example, I have an acquaintance who works in an animation studio (animators are a little different but..) each team was able to decorate their cubicles in elaborate themes, and people in the office would say that they are going to the jungle or the space station. The team very much enjoyed choosing which theme their area would be, and the decoration accumulated so slowly that little working time was lost.
5) Simply say hello to everyone. More then just a passing "hello", stop and genuinely ask how they are doing. If you cycle through everyone on your team, it will be a good reminder to them who are your direct reports.
6) More food: A box or two of doughnuts can go a long way.
Best of luck!