My feeling, in the limited context of mathematics in France, is that the research record is by far the most important thing. Of course, one should do its teaching duty carefully, but all other aspects are tertiary.

I would only mitigate this in favor of the network issue: good research needs to be shown in order to give you benefit, so you should give talks whenever given the opportunity, and in some circumstances propose one. Also, mostly if you advisor does not network for you, you should seize the opportunities to collaborate with more advanced researchers.

I would definitely advise against spending time on administrative things as grant funding and the like. Participate in a collective grant if offered, but that's not a PhD job to do the paperwork.

My feeling, in the limited context of mathematics in France, is that the research record is by far the most important thing. Of course, one should do its teaching duty carefully (edit because it is one's duty, but also because being known as a very sloppy teacher can close some doors), but all other aspects are tertiary.

[Paragraph edited upon further reflexion] I would only mitigate this in favor of the visibility issue: good research needs to be shown in order to give you benefit, so you should have an up-to-date web page, post your preprints on the arXiv, give talks whenever given the opportunity, and in some circumstances propose one. Also some networking may be needed, in particular if you advisor does not network for you. This means for example seizing the opportunities to collaborate with more advanced researchers, send by e-mail your best work to a few people who you think might be interested, etc. Don't overdo it though.

I would definitely advise against spending time on administrative things as grant funding and the like. Participate in a collective grant if offered, but that's not a PhD job to do the paperwork.