OK - realize this is general and there are architectural oddities out there that this would not apply to - but generally, if you have roof or floor rafters, joists or trusses resting on the wall, it is load bearing. IF it is on the (typically) narrow "end" of the house where the roof overhangs (typically) with a few flat 2x4 or 2x6 supports under the sheathing) but no joists or rafter tails sticking out over it, then that is a probably a non-load bearing wall, or carrying only the load of one joist bay. Simple flat roofs and one-way peaked roof houses typically have the two long sides load bearing, the narrower ends not - but houses with full gabled roof (slopes all four ways) typically have roof loads bearing on all four walls. Also, ALL exterior walls are load-bearing to at least some extent if there is another story (not including unfinished attic) above them.
A non-load bearing or lightly loaded wall can fairly readily be penetrated, with typically 10-14" deep beam across the gap up to about 12-14 feet without intermediate columns, longer with intermediate columns - allthe way up to full-side glass curtain walls if desired. Of course, if the floor the columns are on is not the bottom floor (another floor or basement or crawl space underneath) then it may be necessary to beef up the supports in the wall under it somewhat. As Jefferson said, can run from a bit less than a thousand in a simple case to maybe $1500 range typically - not including the architectural finishs for and around the new opening or repair of drywall removed to install underlying supports.
IF load bearing, same story but you need a substantial beam overhead to transfer the load, and if it gets to over about 8 feet wide then you are normally into engineered heavy duty structural beams or steel beams, or need intermediate columns or narrow wall segments to carry the load down to the foundation - and substantial beefing up of underlying walls. Typically about $2-5,000 range depending on whether there are walls below the floor you are working on, and width of opening of course. In seismic/ hurricane/ tornado/ high wind areas can run even higher than that for cases where you are taking out more than about half the total wall width, as special wind/sway bracing has to be added in.
Generally, once you get beyond about 8' wide, it will be cheaper to leave in small wall segments or columns as intermediate supports than to use a wide span - cost roughly increases proportional to the square of the open span, so a 16' opening might be closer to 4 times the cost of an 8' rather than the double you would expect just from dimensions. One way you can eliminate columns if you don't want them is using structural plywood panel wall segments formed in arches to carry more load than a normal beam of same arch top dimension.
Generally speaking, in almost all cost areas, cutting more than a normal mandoor sized opening in a wall requires plans from civil/structural engineer before you will be allowed a building permit - and do NOT slough off on this - I have worked probbly a dozen jobs where contractors jumping in and just doing it resulted in significant structural damage to a building, that then required $10K or more in engineering and repairs to bring back to functional and safe condition.