I am going to Spain in September under a long-term youth mobility visa, which allows me to work, study, and reside in Spain. I am unsure about the travel limitations for this, because my visa is for one year and I want to travel around Europe from Spain for the whole year. Is this allowed?
I'm slightly surprised this isn't a duplicate, but I can't find this question already answered here. EU Regulation 265/2010 amended article 21 paragraph 1 of the Schengen Convention to read
Aliens who hold valid residence permits issued by one of the Member States may, on the basis of that permit and a valid travel document, move freely for up to three months in any six-month period within the territories of the other Member States, provided that they fulfil the entry conditions referred to in Article 5(1)(a), (c) and (e) of Regulation (EC) No 562/2006 of the European Parliament and of the Council of 15 March 2006 establishing a Community Code on the rules governing the movement of persons across borders (Schengen Borders Code) (20) and are not on the national list of alerts of the Member State concerned.
and inserted paragraph 2a, that reads
The right of free movement laid down in paragraph 1 shall also apply to aliens who hold a valid long-stay visa issued by one of the Member States as provided for in Article 18.
So it seems to me that yes, you can use your Spanish long-stay visa as a Schengen tourist visa, and visit other countries in the Schengen area, as long as you don't exceed 90 days outside Spain within any given 180-day period. Since you'll presumably be living and working in Spain, and using it as a base for sorties into the rest of Europe, this should be an easy enough requirement to fulfil.
Note that this applies only to the Schengen area. To visit other European countries, eg the UK, you would have to satisfy their entry requirements. You don't give your citizenship, so we cannot advise you on those.