It'll be easiest, particularly for a starting GM, to use material written for 5e.
But you can certainly convert--for that I'd recommend 2e (see below).
Here are some 5e possibilities, along with sources:
Hardcover 5e books (amazon.co.uk, any other bookseller)
WotC has released three separate seasons' hardcover campaign books.
- Season 1: Tyranny of Dragons, split across two books: Hoard of the Dragon Queen and Rise of Tiamat.
- Season 2: Elemental Evil, in one book: Princes of the Apocalypse.
- Season 3: Rage of Demons, in one book: Out of the Abyss.
- Season 4: one book: Curse of Strahd.
- Season 5: (coming Fall 2016) one book: Storm King's Thunder.
I've played one and am running one currently. I've been happy, as there's more than enough in each to make the basis of a very rich campaign. I've heard some complaints about a linear feel; those complaints have only come from those playing in Adventurers' League, WotC's public-play organization. (Which, in my opinion, tends to create some "this isn't my campaign" problems.)
5e modules (dmsguild.com, drivethrurpg.com)
One source is directly from WotC: they release many downloadable modules designed to dovetail with the storylines presented in the Adventurer's League seasons' hardcovers. (But there's no necessity to ever see the hardcover content--synergy would just enrich each.) To access these you'd need to go through DM's Guild and purchase the modules written for each season\$^1\$ There were fourteen written for season 1; sixteen written for each of seasons 2 and 3, 11 written for season 4. (They're named DDALXX-YY where XX is the season number, YY is the module number.\$^2\$)
Alternatively, plenty have been written for 5e or to be unofficially "5e compatible." You can look at drivethroughrpg.com for official 5e modules or for unofficial ones.
Note that the 5e filter on dtrpg also grabs "D&DNext" modules; I don't know how well those play in 5e. (D&DNext was the name used during playtesting, when rules may have been different.)\$^3\$
ENWorld also had a 5e writing contest that produced good modules. And it's a British site!
Converting old modules (your attic)
If you're going to go the route of old modules I most-highly suggest 2e. I've done 1e, 2e, and 3.5, finding 2e to be the easiest. (As long as you're good at subtracting from 20!) In my experience 2e modules run as-is with no more ad-libbing than I'd exercise with any module written for the system. That is, I don't prep a 2e module any differently than a 5e module: prep is all in understanding the story and personalities; I do the conversion on-the-fly.
\${}^1\$-These modules were previously freely-downloadable through Adventurer's League. Ca. 1/2016 All of this content was moved over to DMsGuild on a pay-basis.
\${}^2\$-The modules don't necessarily go in any order, and they're thoroughly mixed as to which tiers should play them. That is, don't expect modules 1-4 to be the low-level ones and modules 5-9 to be mid-level. Module 4 might be a L1-4 while module 3 was an L11-14.
\${}^3\$-Since this was originally written I have run a few D&DNext-written modules. The monster stats require much love. YOu can either run them as-is but recalculate CRs, or swap out the monsters in the module with ones from the MM. Not too hard, but I found it was something I needed to do ahead of time in order to understand how hard encounters would be, rather than on-the-fly.
