Drill bit Set recommendations

It will only be surface rust. Clean them with vinegar and lightly oil them for protection

Try cleanin up your existing bits with acid first, as Tristan suggest. An SDS drill with SDS bits is best for tough to drill masonry, although no good for wood. (edited)

If you want a decent set that will last forever get a set of Cobalt ones for wood/metal e.g.

Sealey AK4702 Cobalt Drill Bit Set 25pc Metric

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I paid £60 with one of eBay's semi-regular 20% off codes.

For masonry get DeWalt, you should only need 5,6,8mm say, get the ones with "fluted shaft".

Bosch Expert Multi-Material.
Available with straight shanks or hex.

For a diyer just buy any branded set, they are cheap as chips especially if you buy them when they are on offer (look on hukd), avoid the unbranded/ pound shop offerings.
When they wear out , go dull or don't cut properly get them sharpened or simply replace them. A new drill branded bit will out perform a dull worn premium drill bit.
If you are drilling tough masonry or concrete invest in a sds drill.