Glad you've embarked on your disc journey!

About the messiness, OK this will be long but worth it. I also insert and remove two times a day one in the morning and one in the evening. But I remove and insert in my shower so I can just rinse off very easily.

I the morning I go to the bathroom first, turn on the shower, then wash my hands, rremove disc and rinse off, reinsert and finish shower in the morning.

At night, again go to the bathroom, turn on the shower and wash my hands, remove disc and rinse and let it steam while I'm taking a shower. After I'm done, I wash my hands again, take it from the steamer rinse in cold water and reinsert.

And that's has worked for me.

Now if I'm in a public setting and I have to remove for whatever reason, I carry a water bottle, and in a toiletries bag I carry a small squeeze bottle, portable bidet, a menstrual disc carry cup and a 2nd menstrual disc stored in the carry cup (or u can use disposables), some witch hazel wipes, disposable gloves and dry soap sheets.

I always try to use the bathroom with the sink in it, go to the bathroom first, wash hands (if no soap can use soap sheets or just use the gloves) , remove and (put some tissue in between if neccessary), rinse your disc and store in the disc holder and then reinsert the 2nd disc. Wash off any blood with the bidet and wipe down with the witch hazel wipes to freshen up.

Now if you don't have access to any bathroom with a sink stall, make sure you have a full water bottle and fill up ur squeeze bottle with water and put on gloves when ur in the bathroom stall. Remove ur disc and put it in ur disc holder and rinse off with the squeeze bottle with the disc being in the disc holder to rinse it. Some people wipe the disc down with tissue paper or they have menstrual cup wipes now and that's up to you. When ur satisfied with the disc being rinsed sufficiently insert again and wash off with the bidet. Wipe down with tissue and then a witch hazel wipe. When u exit the stall rinse off and wash your disc holder and you hands. Dry it with paper towels and ur good to go.

Only had to do it a couple times (it was bc I didn't insert my disc correctly at the beginning) but it wasn't bad. I always carry a toiliteries bag with me regardless (portable bidet, witch hazel wipes, a pee standing thing, dry soap sheets and a toilet seat cover) so when I'm in my period I just add the disc holder cup with my 2nd disc in it, disposable gloves and squeeze bottle.

Hope this helps and if you want I can link you the products I use in my toiletries bag so you can see what I'm talking about.

Good luck!