Is there an advantage in keeping magazines like "Grognak the Barbarian" for later use? Or can I just get the +1 bonus now without any downside?
When you pick up one of these items (whether that be an Astoundingly Awesome, a Tumblers Today or issue of Unstoppable, etc) it does the "legendary pickup" screen where you get an overview of the item as you pick it up. This is sufficient enough to activate the bonus that you get from that item and this bonus will be listed on the Perks page on your Pip Boy (under STAT > PERKS). You need no further interaction with these items once you've got the perk.
If you pick up multiple magazines in a single pick up, the "legendary pickup" screen will only show for the first one you picked up but all of them will be classed as read.
The magazines give a permanent bonus in Fallout 4 compared to the temporary bonuses they gave in New Vegas. I would suggest picking them up whenever you find them.
When you select the magazine in your inventory, you get an option called "Read". You do not have to do this to gain the effect, it just shows you a close-up of the cover and stays in your inventory.
For the purposes of getting the perk, the magazine is "read" when you pick it up.
Quote from source
Magazines grant the player additional perks when read, or unlock certain benefits. These perks can have multiple ranks, with the rank increasing each time the player reads a copy of the magazine. The magazine may be sold or dropped after being read, and the perk or benefit will be retained.