Marcello wrote a number of sonatas; I suggest the G major for you to start with. Vivaldi wrote 6 for the cello that we bassists stole a long time ago. They usually don't go into the thumb positions, and my students start there after they finish The Jolly Dutchman, The Elephant, etc.
Another excellent solo is the Sonata by Eccles; expressive, challenging, but playable, and we all learn it. The Chanson Triste and Waltz by Koussevitsky are good, too.
A wonderful collection of bass solos with piano accompaniment is entitled Solos For The Double Bass Player, by Oscar Zimmerman and published by Schirmer. I've been playing out of that book for over 40 years.
As for the bass concertos, I'd start with the Capuzzi. It's very playable. Then, I'd tackle the Koussevitsky. It ain't easy but it's very musical and expressive. I never played the Vanhal but it's very well known. I like the Bottesini #2 in A minor; it's a really good one and quite difficult. Good luck and check back!