The main challenge when working on trade in services (and with services in general) is accessing data. Here an (ongoing updating) list of datasets I am aware of/ I have used. If you have any suggestions / questions please get in touch.
Country - level
WIOD: Input-Output tables for 43 countries and 56 sectors (31 within services industry).
SERVICES TRADE RESTRICTIVENESS INDEX (OECD): Restriction to trade in services by type of service and mode of supply.
SERVICES TRADE RESTRICTIVENESS INDEX (WTO/WB): Restriction to trade in services by type of service and mode of supply.
Firm - level
MIDI (Bundesbank): German firms' affiliates in over 180 countries. Access to the data is restricted.