Aligning the answer strictly for Kilimanjaro, I'd first recommend you to select the route that you are going to take. If you want to take the easiest way up, the Marangu route shall be perfectly suitable for you (No offence, just in case if you remain unprepared, if you have been preparing well, you might try the other routes.
I'd consider taking two different routes: One for ascent and the other for descent). You'd probably wanna take care of permits required, if there are any.
Once you are on field there, the thing you must be watchful for is Acclimatization.
This little part "how to get started with something like this" makes me add the below part to the answer.
As far preparing for the higher scale of mountaineering goes, as you predominantly mentioned that you have been doing Day hiking, it should get easier.
1. Strength building and Load Carrying:
Mountaineering doesn’t require outright power-lifting skills, nor it requires a well-gymmed body, or weight-lifting ability, but it does require a fair bit of strength in limbs and shoulders. You’re not just climbing the mountain, but also also probably hauling a large and heavy pack on your back and your body needs to be able to move with that extra weight. As you have been doing day hikes, as assume you are aware of how to walk the mountains. You'll need to practice carrying much a bigger backpack, and still be able to cover the distance. For that you'll need stamina.
2. Increasing the stamina:
Unlike one day hikes, you are not going to scale down a mountain in an hour or two, so increasing your stamina is a huge piece of training for mountain climbing. Being able to keep moving and alert for hours is a key skill for mountaineers. Exercises such as stair climbing (may be with a pack?) and hiking steep terrain are great for working those lower body parts. Even Cycling and Swimming helps to develop stamina.
3. List down the things to pack and way they are packed: I suggest preparing a list of the things that you do not want to miss out when you are on the mountain. There is wide range of the things and in initial days I used to forget on or the other. Its always rightly said that, "I'd always prefer to carry a thing that I may not use up there, but I would rather not prefer not having it when I need it".
e.g. A Flashlight on a day hike? We always plan to finish the activity before sunset, we most of the time do. What if we don't?
So, list down the things, and pack them as you need them. Things which you may need more frequently, you should have them packed in a manner that they are easily accessible.
The bulky and light stuff should go at the bottom of the pack. This question here covers almost every bit about packing the backpack.
4. Get acquainted with necessary Gear and Equipment:
Firstly, You'll need a good backpack to put all the things into it. Get one that suits you. Get acquainted with the terms and the actual equipment that you may have to/want to use. For Kilimanjaro I think you may not need much of a equipment. Just a good backpack, a tent, a sleeping pad and a sleeping bag, a good footwear along with the regular stuff from your Day Hike packs will be more than enough.
5. Get Mad about it to get better:
On a bad day at a Day hike you can be a bit harsh on you and keep going, and at the end of the day you'll find yourself at home safe and sound. In multiday hikes, thats not the case. After a bad day, you may have to still keep going so that things don't get worse. At the most, you'll have a night in the tent or a mountain hut to recuperate in order to either turn back or go with the plan. You gotta fight tough times. And,
Practice and Consistency are key to sustain mountaineering habits. It's always easy to fall off the grid and loose the shape and stop doing the things. It's rather difficult to keep going, keeping the hunger to go on and climb bigger mountains. For me the greediness to trek as much as can keeps me doing the stuff, and sideways, keep doing the stuff prepares me for bigger stuff. Just pat your back and keep it going! There are always new challenges to scale down.
Have a good safe one!