I would start with easy climbs using the gear you got and figure out what you need as you go. Most of my mountaineering gear is also my backpacking gear. Warm down jacket is important for the bigger mountains and nice for the smaller mountains.
My little list would be
- Synthetic base layers (t-shirt & long sleeve)
- Water proof jacket (light weight can work)
- Wind/Water proof pants
- Long johns
- GOOD BOOTS
- GOOD Socks
Sounds like your on the right track. Cheap synthetic clothes are easy to find at second hand stores. Doesn't have to be pretty or expensive, it only has to work.