What do each of the different starting effects do?

The five effect items and what they do are:

• Red Shard: Restores HP. Consumable.
• Stone Sellsword: Offer it to a sanctuary to enables co-op functionality, such as summoning other players to your game.
• Amber Idol: Upgrade material, which is required for creating weapons from boss and enemy remains. One time use.
• Grasping Ring: Increase salt dropped by enemies by 10% when equipped
• Crystal Sphere: Converts a sanctuary to your chosen creed. One time use.

With the exception of the Red Shard, which is complete waste of a gift (very small heal, you and find and buy them everywhere, and you get a much stronger heal item which restocks itself whenever you rest), you can't go wrong with any of the gifts. It really just depends what you're looking to do.

The Stone Sellsword is a good purchase if you're looking to do a lot of co-op, but if you're not, then it's not worth picking up.

The Amber Idol will allow you to transmute a weapon early on, which is pretty handy, but if none of the early weapons interest you then you won't get much use out of it.

The Crystal Sphere will peacefully convert a sanctuary to your chosen creed. You can convert sanctuaries via violence, but it's much easier to just use one of these. If you're looking to join one of the late game (like Sky and Sanctuary) these are really handy, since it can allow you to convert whatever primary sanctuaries you have to your new creed (you can only use Sanctuary NPCs if you're of the corresponding creed). In my experience these items are hard to come by, and I don't think Merchants sell them unless you've got a high creed level, so picking this as your starting gift can be a real help.

With that said, I found the creed you played as didn't really make a big difference. Until you get creeds to the higher levels they're basically all identical, and leveling creeds up is a huge pain. The reason for this is because you need 3 copies of enemy "parts" to level them up, and the drop rates in this game for enemy "parts" are extremely low. So you'll either need to be really lucky or grind a lot to level those creeds up. However, if there is a specific creed you need to join and level up (there are a couple spells you can only get from specific higher level creeds), then this item is a solid choice.

So tl;dr the Crystal Sphere is useful if you need a high level in a specific (non-starting) creed, but not worth taking otherwise.

The Grasping Ring is handy to have right out of the gate since you'll be leveling up 10% faster. You've got four ring slots, so until you're few areas into the game you'll have plenty of space to wear this ring in addition to more useful ones. I consider this to be the "default" gift - if you don't need any of the other gifts, then this is the one you should go with.