It's definitely a long game, we started basically as soon as she had control of her limbs, just including her in every single aspect of life at home and not falling into the trap of nap time = chores, so she was 100% used to all this stuff going on around her. Keeping tasks suitable to her skill level, IE can you find all the socks in this basket for me, and not fussing too much about how much extra mess was made on the way, so if all the laundry went on the floor before it went in the washer, who cares. And then just praise praise praise praise like a loony. Toddlers want to please you! They want to feel like they've done a good job, they want to feel useful and valuable. We also (the adults) made a point of praising each other and thanking each other for small tasks in front of her, so that she got a sense of us all being on a team.
Now that the baby is here, we talk to the little one a lot about how helpful and clever her sister is, and I make sure that there is always room for my big girl on my lap when I'm feeding the baby, so she never feels pushed out. We've worked really hard on making sure the baby is happy to lie by herself for 20min or so, and both parents make sure to give the toddler 1-1 time every day that isn't focused on tasks/transitions.
Setting your expectations is the most important thing. It's not realistic for you to expect your toddler to get everything you need for a nappy change, but that's not really what you're after, what you want is a toddler who feels confident and comfortable in her own space and her own abilities to manage by herself for a bit. I think kids who are particularly clingy are often so because they feel helpless without their caregiver. If you give them the tools to feel like they can manage without you (even though they absolutely can't, cos they're 2) then they're less likely to feel anxious and upset if you need to be focusing on something else for an hour.
Caveat to all this is that I've only done this with the one child and a lot of it may be temperament dependent. DD is very sensitive and has a lot of big feelings, tantrum are off the scale and appear out of nowhere, meal times are very very beige, and I am in no way raising a little angel, but she is comfortable in her own space and I think that makes our lives a lot easier.