What is the best kind of container to carry the litter in? Where should I dispose the litter if I do not have a vehicle?

That depends on how far you go and how much litter you're finding. A five-gallon bucket can be a good choice. You can often get empty paint buckets for free. A sturdy bag is another choice, you can fold them compactly and carry them with you when you go out. It may be easiest to dispose of one or two small items in a public litter bin, if one is available. Parks, shopping centers, and transit stops are good places to look. For larger quantities, contact your city or local government and ask whether they'll send someone to collect bags. Some cities offer litter cleanup tools to volunteers, too.