In BS, there are only two options for gifting: food or parts to build ovens. For food, the options are 150 Chocolate Cake, 200 Brownies, 100 Croissant, 50 Cappuccino or 100 Hot Chocolate. On iOS they're all located about 45-50 pages into the options, after the oven part called Oven Element. Now, if you're going to gift parts, the newest ovens are the goal ovens and their parts are located at the very beginning of the gifting list. This time they're called Cinnamon Knob, Chocolate Gear, Campfire Wood and Marshmallow Knob. It will take you a while to figure out what is a standard part (screw, knob, oven element, ie the parts used in many different ovens) and what is a specialty part that is only used in one oven. I think there's a thread floating around that's dedicated to parts and what they're used for.
I've found that players can be very finicky about gifting so my wall clearly states that I only gift parts, during goals I gift the new parts and I gift standard parts the rest of the time. I also mass gift all neighbors the same gift each day (due to time constraints) which is also on my wall so that players know what I gift before becoming my neighbor.
In general, newer players want/need food gifts and older players have enough ovens and stocked foods that they're in need of goal parts. My advice would be to go through your neighbor list, read their walls and see what sort of things they're asking for. If you've got the time to individually gift, have at it! Lots of players will post which item they specifically want on their walls. If not, see where you're at in the game and what sort of gifts you want/need and try to have neighbors that are at the same point in the game who want/need the same things. This way there isn't any upset at gifting the 'wrong' item as can sometimes happen. The add me threads for BS have several specific to gifting; Add Me if you only want and gift food, Add Me if you only gift parts, you get the idea.
I think that's about everything I can think of in this introduction of gifting, hope I covered the basics and this is at all helpful to you. I realized once I started writing this reply that it is something that could be better explained, in game. They could have info on the back of the gifting options explaining what they're used for so newer players wouldn't have to find second hand sources but apparently they thought this unnecessary. It's a very different system to CS/DS but I think that's because in BS/RS it's all about the ovens. Once you get the hang of it, I'm sure it will become second nature. Good luck!