How to imply “to“ to 2 verbs?

There is a potential difference, in that if you were to include 'both' in your sentence, the repetition of 'to' would not be optional:

I'm eager both to improve myself and to be involved in r&d activities.

Hence the repetition of 'to' suggest that you are referring to more than one thing about which you are eager.

While your second sentence might omit 'to' for no other reason than economy, it could also be taken to suggest that your meaning is similar to the following:

I'm eager to improve myself by being involved in r&d activities.

This is not a steadfast rule, but the second sentence is open to such an interpretation. It's like saying 'I like bread and butter' (together), as opposed to saying 'I like both bread and butter' (but not necessarily together).

• Next Is there real connection between The Machinist and Dostoyevsky's books?
• Previous meaning - Difference between "inflammable" and "flammable"