Rings are full of significance. They represent the never-ending circle of your love for each other, and they are the outward symbol of your invisible bond.
Sometimes rings have special significance. My own very simple wedding band was given to my grandmother on her wedding day over 100 years ago. I've seen rings that were crafted from a rock collected on a special holiday, and a ring which the groom's father placed on his mother's finger decades previously. So, if there is something special about your rings that you would like to mentioned in your ceremony, talk to your celebrant about it.

"Oh, just shove it on..."
This is what one memorable bride said after her beloved had spent a minute or so trying very gently and tenderly to place the ring on her finger... Hilarious and it gave the couple one of many memorable moments in their ceremony.
If you want the rings business to go smoothly, here's a couple of thoughts:
Should you wear your engagement ring on your ring finger? That’s up to you. Lots of people wear the engagement ring on the third finger on their right hand on their wedding day. And some keep it in a safe place and wear it later.
If you are exchanging two wedding rings, think about putting them both in the same box for the ceremony - this saves faffing about with two containers when the ring bearer presents them to you in the ceremony
Traditionally in the UK, the rings are worn on the third finger of the left hand. However, if you wish to wear it differently, that’s absolutely fine – just let your celebrant know so that they give you the correct guidance during the ceremony

Photo courtesy of Raven Photography
To avoid a “Four Weddings and a Funeral moment”, double-, triple- and quadruple-check on the day that the right person has the rings!
Practise placing the rings on each other’s fingers beforehand
Maybe use a little moisturiser on your hands before the ceremony to help the rings glide on
If it's stubborn, give the ring a gentle twist as you place it on - that sometimes helps
If you're getting really stuck, help each other out - and don't panic!
And, if all else fails, instruct your beloved to just shove it on...
Remember, if it's tricky getting the ring on, it's never gonna come off!

I always ask the person giving the ring to place it on the finger, hold it half way down and then repeat some words. And then place the ring on fully. Doing it this way enables you to look into each other's eyes as you say those meaningful words in this important ritual. And it also lets your photographer get into a really good position to capture that moment forever.
A quick word about ring bearers:
The person conducting the ceremony should chat to the ring bearer beforehand to show them where to stand, give them their cue and explain what to do, etc.
Ask the ring bearer just to open the box and hold it out for you each to take the other’s ring (they don’t actually handle the rings as this can lead to them being dropped...)
Ring bearers don’t have to be adults - they can be little people (with a bit of guidance from someone else, if they need it), or they can have tails (probably best to avoid having a cat as a ring bearer though...) – just make sure that anyone involved in supporting, guiding or wrangling the ring bearer knows what to do and when
Practise removing the rings if they are in a special case, or tied to a collar, etc.
And if the box is a bit special (say, it lights up when it's opened), encourage the ring bearer to turn round to your guests to let them see.