Love thy neighbour as you love thyself --> Ama a tu prójimo como a tí mismo (here we don't use the "thy/thee" thing when quoting from the Bible)
New broom sweeps clean --> Toda escoba nueva barre bien.
No sooner said than done --> Not sure about what it means in your homeland, but here we have two, regarding say <-> do + speed: "Es más fácil decirlo que hacerlo" (saying is easier than doing), and "más rápido que inmediatamente" (even faster than 'right away'
)
Out of sight out of mind --> Ojos que no ven, corazón que no siente (literally, "if the eyes cannot see, the heart cannot feel"... or something alike)
Rome was not build in a day --> Roma no se construyó en un día. We also have: "Dios no creó el mundo en un día", which means "God did not create the world in one day"
Slow but sure --> Lento, pero seguro. Also: "Sin prisa pero sin pausa" (no hurry but no break), and "despacito y buena letra" (slow and neatly written, meaning that things must be done slowly in order to have them well done)
When in Rome, do as Romans do --> En la tierra que fueres, haz lo que vieres. It's bad Spanish, and it means: in the land you were to, do what you see (I told you it was bad Spanish...)
