building off of what renovator said, if you have load bearing walls on the first floor, you need to align structure below them in the basement. This can be beams or walls. Frame walls will be your cheapest option in most cases. However if you do not want a wall there, then price is not always the driving factor. However if you are a single story plus a basement, you shouldn't have too many first floor interior bearing walls. If you use steel or LVL beams, they must be sized to carry the first floor load as well as the loads placed on the first floor (2nd floor and possibly roof loads). As you can guess, this can make wood beams, even built up, quite deep. Steel might be your best bet here if you have longer spans between columns in your basement to keep head room heights at ideal levels. I tend to disagree with the claim in wood costing more then steel. Steel will cost more then wood in most cases. The post can be hidden inside walls (why would you do this? save the money and make the wall bearing! unless of course you have a very high point load) or they can be boxed out and made decorative. I joists should actually be cheaper then a floor truss, but typically cost a little more then solid 2x8-12 framing. However IMO, I joists are far superior. They can be had in very long spans so they can lay a continuous 30'+ joist across your load bearing beams/walls and allow for straighter and faster construction. Also, depending on the depth of them and the spacing, they can easily clear span 20'+. Plus they are a "greener" product. Trusses will cost more, but the benefit you have there is the web openings to allow easy smaller duct runs and electrical access.

Structural engineer time. In short... it depends. In our case where we opened up about 24' that used to be the rear wall of our 1 story house there was a column needed due to a point load, a pier and footer added below the column to take the load to the ground, and four LVLs (meeting in the column / side by side) as headers. Both ends of the opening were above existing piers that were deemed to be of more than adequate strength so they were left alone. Then of course there were the materials and fasteners used to make the connections. That said, it took the framers one day to remove the old wall and frame in the new opening, but I already removed all of the drywall, electrical and plumbing ahead of time.

Paul Vla Sen Ko: Double some 2x6s and drive plumbed 2x4s under them to hold up each side of the ceiling joists. Demolish the headers and framing and install (2) 12"x15' Laminated Veneer Lumber members fastened together with a heavy nailing schedule and construction adhesive. Make sure the ends bear on framing that transfers the load to the foundation. Around $300.00 in materials, but you can probably get a refund on the 2x6s if they weren't cut and just have nail holes. I'd take me a day with an unskilled helper. Of course, follow all the previous advice and get this approved by your local division of inspection.

Josh it really depends on the space. To use the example of my personal home... on the first floor we already had one beam below the ceiling that was real, so on the main level it made sense that the two additional beams would be below the ceiling to create a rhythm that subtly defined the spaces. In my bedroom (formerly two incredibly tiny bedrooms and a bath) we wanted it to feel like one room so we had the beams installed above the ceiling plane. In your case based on one photo I would say above the ceiling...