Briefcases, suitcases and bags
Steed's had to travel during The Avengers, either when he is going on a mission as in The Big Thinker,The Town of No Return, The Master Minds he arrives at the island in TNA The Eagle's Nest. Also when he has been invited to accompany Mrs. Peel, or when he's going on holidays such as in Stay Tuned or The Girl from Auntie.

Steed own several suitcases and handbags.

In The Town of No Return and Too Many Christmas Trees Steed packed his shirts and socks in a light brown leather suitcase with his initials "J.S." on it. I guess the suitcase from The Master Minds that Mrs. Peel unpacks is the same one.
In Stay Tuned Steed used two black leather suitcases of different sizes. Tara unpacked it searching for evidence where Steed had been for three weeks.
Bags can be seen in Esprit de Corps, when Steed is doing the laundry and in The Big Thinker or The Bird Who Know Too Much
The town of no return Too many Christmas Trees Stay tuned Stay tuned
Carnations and FlowersHomeProps from the series
TheTown of no Return Esprit de Corps - It could be Mrs Gale's bag The BigThinker The Bird Who Know Too Much The Morning After
John Steed is also using briefcases in several episodes, for example in Trojan Horse, Mr Teddy Bear,  or TNA The Eagle's Nest.