The big 3 in recording software are Emagic Logic audio platinum which I use,Cakewalk Sonar,and Cubase VST (now Cubase SX).A lot of pro studios use Pro tools a hybrid software hardware system that is exorbitantly expensive.There is also a program called Samplitude(recently bought over by Magix) which is superb and a cheaper but good prog called Cool edit pro by Syntrillium.
Any of these progs will do what you need with many features that you may not need at the beginning but could very well come in handy at a later date.Logic audio,Cubase,and Cakewalk all started their life as midi only progs later adding their very powerful audio recording and editing features.Pro Tools,Samplitude,and Cool edit all started as audio only progs and later added limited midi functionality.For the Mac there is also a very good prog called Digital Performer it costs about the same as Logic.
There is also a cheap(I think between 50-100 dollars) but apparently quite good shareware prog call Ntrack which is an audio only prog.
Emagic logic audio platinum costs about £600,Cubase about £500,Cakewalk Sonar about £300,Samplitude about £600,Cool edit about £200 and Protools about £10,000 upwards!(expensive).Emagic,Cubase and Cakewalk all do cheaper versions of their progs like logic audio silver or gold,Cubasis etc but all of these progs will enable you to do much more with your recordings than any Tape based recorder ever could.
Cakewalk Sonar,Cool edit pro and Samplitude are probably the easiest progs to learn for a newbie but imo after having used them all extensively(with the exception of Pro tools which I have only used for short periods)Logic Audio is the most powerful(though they're all pretty powerful)and fastest to use prog to use but it does have a few concepts to get your head 'round before you get good use out of it.
All of these progs are very easy to use for basic multi track recording and if you "just want to be able to click a "record" icon, record, and then play it back while I record another track along side it."There is also,now that I remember it a very good prog called Quartz audio master which is freeware which can be got at the following address
Hope this helps