Non lucrative visa

When you arrive in Spain, you only have 30 days to complete the second part of the process - have your fingerprints taken, get your TIE, etc. so I would suggest that you shouldn't travel elsewhere for long in this period.

Once you have your TIE, you are only allowed to leave Spain for a maximum of 10 months in the first 5 years, so you would need to manage this. 

Although 'freedom of movement' is one of the much vaunted rights of being in the EU, you still can only travel to another EU country for a maximum of 90 in 180 days, although it is unclear if this is enforced. The following is an extract from the EU Commission website regarding this:

If your long-stay visa or residence permit has been issued by a Schengen area country, you can travel to another Schengen area country for 90 days per 180 day period. You must:

justify the purpose of your stay;

have sufficient financial resources for your stay and travel back;

not be considered a threat to public policy, public security or public health.

You can also pass through other Schengen area countries on the way to your host country.

To move from one EU country to another for more than 90 days, you will need a long-stay visa or a residence permit for that country. If you wish to work, study or join your family in the second country, you may have to fulfil more conditions.