How does an LLC go about acquiring another LLC.

You need to double check your facts: if the "LLC we want to acquire is sole proprietorship," then it is not an LLC: an LLC or limited liability company, even a single member (owner) LLC, is completely different and sepaarte from a sole proprietorship. An LLC is its own business entity--it's own legal "person," if you would--while a sole proprietorship has no independent existence and is the owner.
If this is an LLC, the owners (called members) can simpy sell it to you for an agreed-upon price: all you need is a contract that for $X they are selling you Whatever-the-Name LLC. It's that simple; and when you buy the LLC, you will own everything it owns. (It's a good idea to include an inventory of what it owns--and therefore what you are getting--as part of the contract to avoid a later disagreement.) Your LLC can be the buyer. Your LLC can then either run the acquired LLC or take what it wants from it and shut down the LLC itself. 
If it is a sole proprietorship, you need the sole proprietor to enter into a contract to sell you all of his or her business assets (e.g. all of the equipment, money in the bank, supplies, accounts receivable, business name, etc.) and to transfer to you ("assign") any of the contracts or leases you want to take over. Since there is no overall business structure to buy, you will need to list each thing you are buying or which is being transferred to you (or your LLC; whomever is purchasing) separately, to make sure it is acquired.
In either event, a contract is what you need, but the big distinction is that in one case (LLC) you get a business structure or entity and everything it owns; in the other, you are buying each thing you want separately.