Silkair is part of Singapore Airlines and their rule book is what applies. One one ticket they will check a bag through for you, on two, unless you have SIA status meaning PPS rather than Krisflyer they will not.

Silkair arrive in T2 at Changi where you would have to clear immigration and customs, collect your bag and move to T1. Immigration is fast in Singapore and at Changi bags are often on the carousel by the time you get there. It's only a short walk to T1 and there are plenty of trolleys available should you need one.

Should you be able to check your bag through, you can move from T2 to T1 very easily, I prefer to walk when in T1, 2 & 3 at Changi - head down past the T2 E gates and when you get past E28 you are in T1. There are travelators and a train too but beware with the train you can at times end up going backwards to get on it especially if you arrive at an E gate.

Changi is probably the most civilised airport in the world!