I already know that there's the Plane Shift spell, but it needs a 7th-level slot that I don't have at 10th level. Are there any other ways — spells, magic items, or something else — that I could use at 10th level to travel to other planes, or a way I could gain access to a 7th level spell slot to cast Plane Shift with?
In Faerûn, yes you can!
On p153 of City of Splendors: Waterdeep is the 5th level spell Halaster's fetch II, which is a conjuration (calling) spell whose description says:
As summon monster II, except called creatures do not disappear at the end of the spell's duration. However, they become free-willed and outside the control of the caller after the spell expires.
On p130-2 of the Planar Handbook is the ur'epona which can plane shift 1/day as a 9th level caster. According to PlH p134, the ur'epona can replace the celestial riding dog on the summon monster II list.
So you can get there with a 5th level spell. That is, if you have access to the PlH, and your DM is willing to allow you to summon an ur'epona, as the PlH suggests.