ask someone to help you with the bags, then lift the buggy on. KEEP THE KIDS IN THE BUGGY UNTILL THE DOORS ARE SHUT, STRAP THEM BACK IN THE STOP BEFORE YOURS!! gather bags together, and be at the doors before the stop.

I just did it - plan!
I would book seats all the way there and back in advance if you can. If you pay for the children you can book seats for them (on Virgin, anyway) and then you get more space at least. If they go free, you can't. You could ask for table seats or those ones with extra space.
We arrived early and I asked the person on the platform who works for the train company to help -will my buggy go in the train door? Where is the best place to leave it?
If you are holding people up they will probably help just to get you out of the way!
I gave my oldest (but she is 5) a bag to carry across her body with her colouring books and pens and a drink and a banana in. I carried a rucksack (hands free!)with the baby's stuff in the top.
Hope this helps.

We packed up and stood by the doors the stop before ours, too.

If buggy fits through doors keep kids in buggy and use a back back. Arrive early and find out where the disabled carridge will stop if possible. Means you have room to get orgainised once your on. Don't rush to get on, If you don't get in at the disabled bit, just get buggy in and push forwards so it sits in front of the opposite doors, then people can get on behind you.
I used trains a lot and was only on for short time so kids stayed in buggy/ pram and I usually stood besdie them in doorway. The newer trains here have loads of space and you can park buggy facing seats.

When you get on face forward and tip up front wheels. When getting off best to go backwards unless very level. I used to have a huge drop to get down and always managed it backwards. Try to find out which side the platform will be at. If you get stuck facing forward just ask for help, as Helsey said if your in the way they will help to get you out of the way.

Give yourself lots of time, find out if doors are single or double and avoid rush hour if you can.

Does your 3 year old really need a buggy?

If I were in your situation, I would take a lightweight umbrella-fold buggy and a sling. The baby would go in the sling and toddler in the buggy when you needed to have the toddler ultra-secure. Then put baby in the buggy and let the toddler walk when you were safe at your destination.

Bags are a problem - obviously, pack as little as possible and use you buggy handles. If you need to take a lot, use a rolling suitcase, but that would mean steering your buggy with one hand.

I have found that other passengers and the staff are really helpful, but I think you have to be able to manage by yourself too, just in case.