Welcome, Igor!
The blue triangle numbers are for Provincial Highways. They are fine for bikes. The ones you can’t go on are Freeways and Expressways. On my map, the Freeways have a kind of ‘cloverleaf’ or flower shape with a yellow background and the number in black.
The Provincial Highways go right through cities and towns including Taipei, but sometimes its easy to lose them. If you are going along a Provincial Highway and it goes into a city, it will have a street name as well. Sometimes in a city the street name takes priority, and road signs don’t have the blue triangle with the road number on, even though the road is both street and highway. For example, the no.3 Provincial Highway runs right through Taichung. At one point it is Chungshan (Zhongshan) Rd, at another point it is Beitun Rd. It doesn’t stop being the no.3 Provincial Highway as well, but there are some bits where the road signs don’t have the blue no. 3, although it’s getting better.
When I was in the hotel in Taipei with my mother and my friend, we spent some time looking at various maps trying to work out the best way to get out onto the no. 3. My friend went back a day earlier than us, and he found that although there are a couple of tricky, confusingly-signed junctions, Hoping (Heping) West Rd basically turns into the no. 3! Based on his advice, my mother and I drove straight out of Taipei on the no.3 highway all the way.
Another provincial highway leading out of Taipei is the no. 2 over Yangmingshan and out to the coast. To get on it, go along Chungshan (Zhongshan) North Road up into the Shilin district. Turn right (onto Yangmingshan Road I think) in the direction of the National Palace Museum. If you just keep on that road you will go up the mountain and will come to a fork in the road. If you keep on that main road you will eventually get to the coast, close to Yeliu, where there are some beautiful rock formations (you have to pay to get in but it’s worth it). If you continue along the no. 2 you will get to Keelung. Just one point- the main no.2 highway actually goes right round the north coast. The bit over Yangmingshan is the number two with a little Chinese character under it, a kind of subsidiary of the main no.2. It’s still a Provincial Highway and still uses the blue triangle signs, though, and I don’t think you’d get lost because the place names are all romanised as well.
I’m not very familiar with Taipei so I hope somebody can make my directions clearer and also tell you of other good roads to go on. Taking the no.9 out southeast of Taipei looks as if it would be a nice ride- I think somebody mentioned it before on this thread.
No problem. A friend of mine recently went over the Southern Cross-Island Highway, much higher and more demanding than the road to Keelung, on his old 125 scooter. I think fellow poster Feiren did the same. You should make sure your bike is in good mechanical condition first, though, including brakes, tyres (Enough tread? Not cracked? Air pressure OK?) and engine (clean spark plug and carburettor, regular oil changes) amongst other things.
No you don’t. Of course you won’t get fantastic speeds or performance out of a 125 engine, but that’s just as well because the scooter brakes and tyres couldn’t handle a great deal of speed anyway. Just take it easy and enjoy the ride. And remember to be aware of what’s happening on the road all around you and try to anticipate tricky situations before they arise.