If your computer is modern at all and has USB or firewire then I would suggest you getting an audio interface with microphone preamps built in. It will eliminate the need (initially) for a mixer, because they basically function as one.
If you are not going to record more than 2 tracks at a time you should be good to go for a while.
i don't know what is available to you so I won't comment on brand names.
As far as Microphones go A dynamic type microphone would work well on Tin Whistle and Bohran/ djembe. It would calm some of the shrillness that some tin whistles have and most have adequate frequency response.
If you go with a condensor type mic so that vocals and acoustic guitar / mandolin type thing come out more pristine...watch out and not get an overly bright / cheap mic because the whistle will take your head off at the ears.