Almost all undergrads have to pay their dues before recieving any independent responsibilities outside feeding animals or washing dishes! So, I would start by finding a PI that is friendly and willing to give you independent opportunities/projects in the future; make sure to discuss this and several other factors before settling on a lab.
Once you are in a lab, you will most likely start by learning basic techniques in the lab and basic experiments. On the side, you may be expected to do some menial dirty work such as washing glassware, etc. However, it is important to impress the postdoc or whoever you are working directly under from the get-go. You should only need to be taught an experiment once--take good notes and relax while performing the actual experiments. You will likely stay with the postdoc or whoever for a while, ranging from a few months to a year, hopefully no more than that!
One crucial task is to read up on the current AND past literature pertaining to your lab. This takes time so start early, build good habits, and read atleast one article every 2 weeks and understand it and elicit the big picture from it. You will likely be assigned several journals to read but if not, take the initiative and impress your PI and postdoc with your knowledge.
By this time, you should be able to work independently in the lab without asking for much assistance from senior lab members. Also, you should ideally have access to the lab via key/card/etc. Now, the best way to demonstrate you can successfully run a project by yourself is to
1) Spend a TON (depends on the lab but ideally 20/25+ hours a week) of time in the lab. Nobody is going to be willing to do the majority of the work for YOUR project, so be prepared to spend nights, weekends, holidays in the lab. After all, don't you want to make progress on your own project? 🙂
2) Suggest a potential project idea to your PI, not your postdoc. Almost without a doubt, your PI will claim he/she has a better idea for a project; however, you show initiative and ability to come up with a plan for a project, a very important quality of a researcher. Now, it is much easier to work together with a PI in coming up with a project for you--one that is small enough to finish with the time allotted as an undergrad but good enough to result in a publication. You need to provide input on the potential project topics in terms of ideas and the proposed steps to the project to demonstrate your capability.
Basically, your PI will assign you a project of your own when you can 1) perform experiments independently and 2) when you can think for yourself without being a robot programmed by the postdoc you are working under.
Cheers 😀