This one is a many-layered discussion because it deals with water. Water can come from your tap, a water fountain, a restaurant, a filter or bottled water. This post will only cover the aspect of transporting your choice of water and not how the water is obtained, except for portable bottles.
Last year Americans used 50 BILLION plastic bottles, that is around 170 bottles per person, per year. Of that 50 billion, 38 BILLION were NOT recycled. That means that only about a quarter of water bottles are recycled. Before you climb into your comfy "but I recycle" blanket, consider that "recycling" water bottles is only a down-cycle, meaning that it is not cyclic - that PET water bottle has this life and one more (turned into park benches, bridges or fiber) before that, too, is sent to the waste heap, where, if it is lucky, will wind up in a landfill instead of the ocean.
People, this is another easy fix! One that will even save you money in the long run. Being ripped off by water bottle manufacturers irks me and it should irk you, too. Buying prepackaged water in bottles is completely unnecessary and, in most cases, isn't even good for you; the practice of filtering the water is questionable at best and belief in a corporation's good word and honesty should be at the top of your conspiracy list.
Conspiracies aside, it is a fact that you do not need to buy a bottle of water - you can take it with you. If you like cold water, get one that is insulated and fill it yourself. Truly, nothing could be easier - except maybe taking your own grocery bags to the market with you.
Your real important question shouldn't be "can I do this?" or "should I do this?" instead it should be "which bottle should I get?"
Insulated or not? If you like it cold, get insulated.
Material? I like to use stainless steel bottles (definitely NOT plastic), but some people like glass ones. If you opt for glass, consider using a mason jar with an attachment like this, from Cuppow.
The lid: this one is important; you should be able to clean the lid thoroughly - and that means nothing fancy and complicated. Use the simplest lid that you can dry out well to avoid mildew buildup.
My family uses Under Armour Vacuum Insulated Stainless Steel bottles. The beauty of this bottle is many: it is stainless steel - no flavor leaks into the water. It is insulated so beverages stay cold. The flip top lid is simple and easy to clean and keep dry. We bought extra lids (about $4 each, including shipping) and we rotate the lids so they can be cleaned and allowed to dry out very well. If you get this bottle, email me and I'll give you the contact info for more lids. The process was a bit of a run-around at the beginning, so I can save you a few emails.
Other bottles to consider:
Klean Kanteen, they also sell insulated bottles and have a whole range of product. The lids are simple enough. They also have a bamboo cap version!
Glass water bottles from ReuseIt.com They have tons and tons of reusable bottles and bags, etc.
You can also purchase a bottle sling - makes carrying the bottles, especially on walks, much easier.
In any case, go search and get yourself a bottle and, more importantly, USE IT!