There are two or three waymarked routes.
One runs straight south from the city centre on the left bank of the river, up-river, crossing over at Roques to Pinsaguel, then on up river via Muret and Carbonne, eventually Saint-Gaudens, eventually joining the Piémont Way at Barbazan.
The other follows the Arles Way out of Toulouse via Colomiers, Pibraq, Léguevin and so on, L'Isle-Jourdain, Auch, eventually Maubourguet ; there is then a waymarked route from Maubourguet to Lourdes, with two variants -- either straight south from Maubourguet up river along the Adour to Tarbes, then either a little DIY along that same up-river route up to the Piémont Way at Bagnères de Bigorre ; or carry along the Arles Way a little further after Maubourguet, then take the GR101 through a lonelier route directly to Lourdes.
The route via Saint-Gaudens would have many advantages at start until Barbazan, i.e. it goes through many small towns and villages as well as being mostly a river valley route. It also gets you quickest out of the suburbs. Saint-Gaudens is also a lovely town BTW. The Piémont Way between Barbazan and Lourdes is more mountainous, but the waymarked track does occasionally lead via excessive detours up and downhill rather than a quicker small country road tarmac option -- though perhaps the mountain hiking would be a plus for you ?
The option up the Adour via Tarbes, after the necessary DIY from Tarbes up to Bagnères, would lead to either a rather mountainous option along the Piémont Way to Lourdes, or to a still mountainous but less so multiple-choice DIY possibility on tarmac to Lourdes via various villages.
The GR101 is the "easy" option, including in its possibilities for tarmac variants via towns and villages.
I might personally go the Tarbes Way then DIY out of there along a cyclists path through Juillan & Louey, through Adé to Lourdes. Which I suppose technically constitutes a fourth waymarked possibility. It's just that Tarbes is actually a fairly nice city, and they have good food and etc. It's quite possible BTW that this would be closest to the "historic" route from Maubourguet & Tarbes to Lourdes, and it would certainly seem the more "sensible" one, all waymarks & "typical" hiker stuff excluded.
The GR101 appears to go out of its way to get people into hills and mountains and away from villages and pleasantness for what I would personally consider as no good pilgrim purpose -- though some parallel tarmac routes into the villages that it tries to avoid could be worth it. But I simply can see no advantage in avoiding Tarbes so extremely as the GR101 does, nor even at all !!