Definitely go conduit here
I would lay two conduits (one for mains, one for data/telecom) if I were in your shoes; this way, you don't have to dig anything up later to upgrade it. 1" is adequate for most telecom cables you'd want to run; however, while a second 1" will accommodate smaller mains feeders (up to 60A, or perhaps a bit more), I would recommend using 1.5" instead if you are interested in having a full panel at the shed with a feeder to it instead of a simple MWBC+disconnect box configuration. That way, you will have enough room in the future for a 100A or 125A feeder that's ample for a full-blown shop, ADU, or other such power-hungry use in the future. Don't forget to leave pull-strings in conduits you aren't going to use right away!
Fiber is also a very good idea, because you'll need some primary protection otherwise
The other thing you'll want to do is get a couple of small switches with SFP support (or media converters, if you can't find suitable switches), a pair of SFP transceivers with duplex LC connectors on them and some preterminated direct bury/wet location duplex fiber patch cord (LC/LC) of the appropriate length for your run (don't forget to leave length for the stub-ups at each end!); this way, you won't have to deal with the problems inherent with running copper telecoms lines outside. If you do insist on using copper for this application, you'll need to fit UL497 listed primary protectors at both ends of the run to satisfy 2017 NEC 800.90(A) regarding interbuilding circuits that are exposed to lightning, as that's the main threat to your Ethernet run, not mains interference.
This requirement, unfortunately, is not easy to find parts for when it comes to datacom circuits. ITW/Linx is the only vendor I've been able to find carrying any (no affiliation with them), and their CAT6-75-110 model would be the best suited for your application I reckon. You'll need one for each end of the link in order to protect both buildings from lightning or power cross induced voltages. Furthermore, you'll need to connect their grounding points to the intersystem building termination or grounding electrode system for their respective buildings with a 10AWG minimum bare copper ground wire in order to meet installation and NEC bonding requirements. This also means the shed will need to have a grounding electrode system (2 8' rods 8' apart, connected to the disconnect or subpanel with 8AWG minimum bare copper), even if it is not otherwise required to have one by Code.