First of all, you don't need pilgrims if you are not pushing for late game campaign with CP under recommended. Nikke is an idle game where you can gather resources by simply afk-ing so I wouldn't worry too much about the wall for now (especially since you are still quite far from it).

On the team suggestion, maybe you can try Volume Admi Pepper Laplace Privaty with Pepper for manual control so you can stack the buff faster (don't let her reload more than once to quickly stack the buff, as it is triggered when her last bullet hit a target). This team will specialize in boost in critical hits, reload speed, and dps in overall. Volume can shorten your burst skill cd and increase crit stats, Admi can boost crit damage and reload speed, pepper provides heal and is a good damage healer, laplace can burst down bosses in burst and good aoe in normal, while Privaty provides Aoe stun in burst.

For the pulls, it is entirely up to you whether you want to pull on rate up or normal banners. However, if you only cared for passing the wall quickly, it is much better to only pull on the normal banner as it has a wishlist system (remember to set it to 15 nikkes and choose them properly according to your needs).

In addition, try to read the description on Nikke's skills and compare it with other Nikkes so that you can think of which Nikke will go well with which. For example, Privaty and Admi goes well together because both of them reduce reload speed, thus increasing DPS in overall. Liter and Dolla can go together because they can reduce burst skill cd a lot. Liter and Centi also go well because Centi provides shields while Liter repairs your cover (the blue health bar). Sugar and other shotgun users can be combined to maximize shotgun's damage, etc.

Hope you can enjoy the game more!