Yes it's a small country but you are pushing it with a week. If you really must do all three, .....Arrive in Sri Lanka and head south to Galle. Day 1. . ...... You have your choice of taxi, bus , or train... but base it on when you get in. If train is available, then go for it...If not, take the bus as they are readily available..They are about every 20 mintutes to Galle. Personally i like the bus going down and then the morning train coming back as it works better with timings IMO, however with the rush that you are in, it might actually work better if you take the bus back up as it leaves much earlier.... ie 6-7am. .....
There is an airport bus from the airport that will drop you directly at the bus station and if the current bus to Galle does not have seats, let it go and take the next one in 1/2 an hour. Its a long way to stand... .... Whatever you take down, take the other back up if that suits your schedule. Day 2 around Galle, Unawatuna, Maybe marissa.... Day3 is a travel day... I persnally like the bus going up the middle via Ella from Matara, but you don't have time for this and this is one of the most packed busses i've ever been on in my life. So back to Colombo by bus and then immediate transfer to Kandy bus . In Kandy end of day 3, so 4 in Kanday , next day to Sigiriya, and then Day 6 or 7 return direct to Colombo depending on flights
Whoo...just exhausted thinking about it. If you are going to miss something, skip the beach and do Kandy/Sigiriya first. A beach is a beach in some ways, but Sigi is special.